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The sphere is a three-dimensional shape with the following properties: - It is perfectly symmetrical and has no edges or vertices. - All points on the surface are equidistant from the center. - It has the smallest surface area for a given volume of any shape. - Spheres appear naturally in bubbles, water drops, and planets like Earth due to this efficient use of space. - It can represent the simplest single point with no dimensions, and also the most complex shape containing all other shapes within it.

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Volume of cylinders

Year 8 Maths activity to calculate the volume of cylinders using photographs of containers and tanks on the farm.

Areas of Plane Figures

The document defines the formulas to calculate the areas of various plane figures including triangles, squares, rectangles, parallelograms, rhombuses, trapezoids, and circles. It provides examples of applying the area formulas to solve problems involving these shapes. The objectives are to learn how to compute areas, determine measurements given areas, complete interactive exercises, and develop problem-solving skills related to area calculations.

Area of sectors & segment ananya

This document discusses the areas of circular sectors and segments. It defines a sector as the region between two radii and an arc, and provides a formula to calculate the area of a sector based on its central angle and radius. A segment is defined as the region between a chord and arc, and the document explains that the area of a segment can be calculated by taking the area of its corresponding sector and subtracting the area of the triangular portion cut off by the chord.

Square and square roots

This document provides an overview of squaring numbers and finding square roots. It discusses key concepts such as:
- Squaring a number means multiplying a number by itself
- Perfect squares are numbers that can be written as the square of a whole number
- The square root of a number is another number that, when multiplied by itself, equals the original number
- Examples are provided of finding the square of numbers and the square roots of perfect squares.

Grade 9 pythagorean theorem

The document discusses the Pythagorean theorem. It defines the terms leg and hypotenuse in a right triangle. The Pythagorean theorem states that the sum of the squares of the legs equals the square of the hypotenuse. The document provides examples of using the Pythagorean theorem to solve for missing lengths in right triangles.

Division of Radicals.pptx

This document discusses dividing radicals and rationalizing radicals. It provides examples of rationalizing radicals by applying the law of radicals to make the denominator free of radicals. Examples of dividing radicals with the same index are also shown. The steps for dividing radicals are to divide the radicands and rationalize if needed to remove radicals from the denominator. An activity is included where students work in groups to solve radical expressions and present their work.

Surface area and volume powerpoint

1) The document provides formulas for calculating the surface area and volume of basic geometric shapes like rectangles, triangles, circles, cylinders, prisms, and triangular prisms.
2) It explains how to identify the key dimensions needed to substitute into the formulas, like finding the radius, diameter, height, and base/length of an object.
3) Examples are provided to demonstrate applying the formulas step-by-step to calculate surface areas and volumes.

Pythagorean Theorem Lesson

The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It defines the key terms hypotenuse, legs, and explains the a^2 + b^2 = c^2 formula. Examples are given to demonstrate applying the theorem to find the unknown side of a right triangle given the other two sides. Practice problems are provided to reinforce the concept.

Volume of cylinders

Year 8 Maths activity to calculate the volume of cylinders using photographs of containers and tanks on the farm.

Areas of Plane Figures

The document defines the formulas to calculate the areas of various plane figures including triangles, squares, rectangles, parallelograms, rhombuses, trapezoids, and circles. It provides examples of applying the area formulas to solve problems involving these shapes. The objectives are to learn how to compute areas, determine measurements given areas, complete interactive exercises, and develop problem-solving skills related to area calculations.

Area of sectors & segment ananya

This document discusses the areas of circular sectors and segments. It defines a sector as the region between two radii and an arc, and provides a formula to calculate the area of a sector based on its central angle and radius. A segment is defined as the region between a chord and arc, and the document explains that the area of a segment can be calculated by taking the area of its corresponding sector and subtracting the area of the triangular portion cut off by the chord.

Square and square roots

This document provides an overview of squaring numbers and finding square roots. It discusses key concepts such as:
- Squaring a number means multiplying a number by itself
- Perfect squares are numbers that can be written as the square of a whole number
- The square root of a number is another number that, when multiplied by itself, equals the original number
- Examples are provided of finding the square of numbers and the square roots of perfect squares.

Grade 9 pythagorean theorem

The document discusses the Pythagorean theorem. It defines the terms leg and hypotenuse in a right triangle. The Pythagorean theorem states that the sum of the squares of the legs equals the square of the hypotenuse. The document provides examples of using the Pythagorean theorem to solve for missing lengths in right triangles.

Division of Radicals.pptx

This document discusses dividing radicals and rationalizing radicals. It provides examples of rationalizing radicals by applying the law of radicals to make the denominator free of radicals. Examples of dividing radicals with the same index are also shown. The steps for dividing radicals are to divide the radicands and rationalize if needed to remove radicals from the denominator. An activity is included where students work in groups to solve radical expressions and present their work.

Surface area and volume powerpoint

1) The document provides formulas for calculating the surface area and volume of basic geometric shapes like rectangles, triangles, circles, cylinders, prisms, and triangular prisms.
2) It explains how to identify the key dimensions needed to substitute into the formulas, like finding the radius, diameter, height, and base/length of an object.
3) Examples are provided to demonstrate applying the formulas step-by-step to calculate surface areas and volumes.

Pythagorean Theorem Lesson

The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It defines the key terms hypotenuse, legs, and explains the a^2 + b^2 = c^2 formula. Examples are given to demonstrate applying the theorem to find the unknown side of a right triangle given the other two sides. Practice problems are provided to reinforce the concept.

Polygons

The document defines and discusses different types of polygons. The main points are:
1. A polygon is a plane figure formed by three or more line segments that intersect only at their endpoints to form a closed region.
2. Polygons can be classified as convex or concave based on whether any line segment connecting two points within the polygon lies entirely inside or outside the polygon.
3. Regular polygons are polygons that are both equilateral (all sides the same length) and equiangular (all interior angles the same measure).

Cones

A cone is a 3D geometric shape that tapers from a flat circular base to a single point called the apex. The formulas for the volume and surface area of a cone involve the radius of the base and the height or slant height. Word problems apply these formulas to calculate missing values like height, volume, or surface area when other values like radius or volume are given.

Area and circumference of circles

This document defines key terms and formulas related to circles, including circumference and area. It defines a circle as all points equidistant from a given center point. The radius is the distance from the center to the edge, and the diameter runs through the center. Circumference is defined as the distance around the circle and is calculated using either C=2πr or C=πd. Area is calculated as A=πr^2. Several examples are provided to demonstrate calculating circumference and area using these formulas.

Similar Triangles

The document discusses similar triangles and scale factors. It provides examples of similar triangles in nature, art, architecture, and mathematics. It explains the different rules to determine if triangles are similar: AAA (angle-angle-angle), PPP (proportional property), PAP (proportional angles property), and RHS (right-hypotenuse-side). Examples are given applying these rules to prove triangles are similar and calculate missing side lengths or scale factors.

Basic geometrical constructions

Basic geometrical constuctions is how to construct angle by using compass and ruler.
this slide can help students or teachers to construct any angles especially for special angles they are 90 degree, 60 degree, 45 degree and 30 degree.

Surface Area and Volume

This document provides information on calculating surface area and volume for various 3D shapes including prisms, cylinders, pyramids and more. It defines key terms like surface area, volume, and includes formulas and examples for finding the surface area and volume of rectangular and triangular prisms, cylinders, and pyramids using measurements of lengths, widths, heights, radii, etc. Practice problems are provided throughout for additional examples of calculating surface area and volume.

Area of a triangle

To be able to calculate the area of a triangle using trigonometry up to Grade A*. By the end of the lesson, all students will be able to identify when to find the area of a triangle using trigonometry. Most students will be able to calculate the area of a triangle using the formula. Some students will be able to calculate GCSE area of triangle problems using trigonometry up to Grade A*. The document then provides examples and practice problems for students to calculate the area, missing sides and missing angles of triangles using trigonometry.

Introduction to Pythagorean Theorem.pptx

This document discusses triangles and the Pythagorean theorem. It defines different types of triangles, including right triangles. It explains how to calculate the area of triangles. The main focus is on the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Several examples are provided to demonstrate how to use the theorem to solve for missing sides of right triangles.

Square roots

This document provides information about square roots and real numbers. It includes:
1) Examples of finding square roots of perfect squares and estimating other square roots.
2) Classifying different types of real numbers such as rational numbers, irrational numbers, integers, natural numbers, and more.
3) Examples of classifying given numbers as rational, irrational, integers, and other categories.

Arcs and Central Angles

Central angles are angles whose vertex is the center of a circle. A central angle separates a circle into two arcs: a minor arc and a major arc. The measure of a minor arc is equal to the measure of its central angle. The measure of a major arc is equal to 360 degrees minus the measure of the minor arc. The measure of an arc formed by two adjacent arcs is the sum of the measures of the individual arcs. If two minor arcs in the same or congruent circles are congruent, then the corresponding chords are also congruent.

Integers and Operation of Integers

This document provides information and examples about integer operations:
- Addition of integers follows the same rules as normal addition, such as 20 + 10 = 30 and -40 + -60 = -100.
- Subtraction of integers is performed similarly to addition, such as -3 - 7 = -10 and 15 - 9 = 6.
- When multiplying integers, the product is positive if the signs are the same and negative if the signs are different, exemplified as -2 × 6 = -12 and 2 × -3 = -6.
- For division of integers, the quotient is positive if the signs are the same and negative if the signs are different, with examples like 12 ÷ -4 =

Circles - An Introduction

THIS POWERPOINT PRESENTATION ON THE TOPIC CIRCLES PROVIDES A BASIC AND INFORMATIVE LOOK OF THE TOPIC
_________________________________________________
LIKE ...COMMENT AND SHARE THIS PRESENTATION
FOLLOW FOR MORE

Zero and Negative Exponents

- The document discusses how computers use negative exponents to process fractions and percentages during photo processing, which allows apps like Photoshop to shrink photos.
- It explains that negative exponents represent fractions, with the item with the negative exponent moving to the denominator when written as a fraction. This allows computers to perform mathematical operations involving fractions.
- Examples are provided of how negative exponents simplify to fractions through the rule of a-m = 1/am, with the item in the exponent moving between the numerator and denominator.

Estimating square roots

To estimate square roots:
- Recall perfect square numbers like 4, 9, 16, 25, etc.
- The square root of a number is between the greatest smaller perfect square number and the smallest greater perfect square number.
- Examples show estimating the square roots of 32, 129, and 875 by identifying the closest perfect squares above and below.

Perimeter, area and volume

This document discusses perimeter, area, and volume. It begins by defining perimeter as the distance around a shape found by adding all the side lengths. It provides examples of calculating perimeters of rectangles, irregular shapes, and converting between units. It then defines area as a measure of how much surface a shape covers. It gives formulas and examples for finding the areas of rectangles, triangles, parallelograms, trapezoids, and irregular shapes. Finally, it discusses surface area as the total area of all faces of a shape. It provides the surface area formulas and worked examples for cuboids and cubes.

Classifying Angles

This document discusses different types of angles including acute, obtuse, right, and straight angles. It defines an angle as being formed by two rays sharing an endpoint called the vertex. Angles are measured in degrees, with acute angles between 0-90 degrees, obtuse angles between 90-180 degrees, right angles equal to 90 degrees, and a straight angle equaling 180 degrees. It includes examples of each type of angle and encourages identifying them in a game.

Parallelogram area

This document discusses how to calculate the area of parallelograms. It defines key terms like base, height, and parallelogram. The area of a parallelogram is calculated as the base multiplied by the perpendicular height. Examples are provided to demonstrate calculating the area or finding the missing value given the area. The document also briefly discusses rhombuses, noting they are a type of parallelogram with four equal sides and their area is calculated the same way.

11 2 arcs and central angles lesson

This document discusses arcs and central angles in circles. It defines arcs as curved lines formed when two sides of a central angle meet at the center of a circle. There are three types of arcs: minor arcs are inside the central angle and measure less than 180 degrees; major arcs are outside the central angle; and semicircles measure 180 degrees. The measure of an arc depends on its type and the measure of the corresponding central angle. Rules are provided for calculating arc measures using central angles and properties of adjacent arcs. Examples demonstrate finding arc measures using these rules and properties of circles.

Area of trapezium

A trapezium is a quadrilateral with one set of parallel sides. To calculate the area of a trapezium, divide it into a triangle and rectangle by connecting the non-parallel sides. The area formula is (a + b)h/2, where a and b are the parallel sides, h is the height or altitude, and (a - b) gives the base of the triangle formed. An example calculates the area of a given trapezium using this formula.

Polygons

Chapter 3 Polygons
3.1 Definition
3.2 Terminology
3.3 Sum Of Interior Angles Of A Polygon
3.4 Sum Of Exterior Angles Of A Polygon
3.5 Diagonals in one vertex of any Polygon
3.6 Diagonals in any vertices of a Polygon
3.7 Quadrilaterals

surface areas and volume

The document discusses the volume and surface area formulas for different geometric shapes including spheres, cones, cuboids, cubes, and hemispheres. It defines a sphere as a perfectly round object where all points are the same distance from the center, and gives its volume formula as 4/3 * π * r^3 and surface area as 4πr^2. It also provides the surface area formulas for cones, cuboids, cubes, and hemispheres. Examples are given to demonstrate calculating surface areas using the appropriate formulas.

MATHS PRESENTATION

The document thanks the teacher, Mrs. Leena Saji, for giving the opportunity to do a presentation. It also thanks parents and friends for their help in finishing the presentation on time. Finally, it thanks God for everything and bringing the presentation to fruition.

Polygons

The document defines and discusses different types of polygons. The main points are:
1. A polygon is a plane figure formed by three or more line segments that intersect only at their endpoints to form a closed region.
2. Polygons can be classified as convex or concave based on whether any line segment connecting two points within the polygon lies entirely inside or outside the polygon.
3. Regular polygons are polygons that are both equilateral (all sides the same length) and equiangular (all interior angles the same measure).

Cones

A cone is a 3D geometric shape that tapers from a flat circular base to a single point called the apex. The formulas for the volume and surface area of a cone involve the radius of the base and the height or slant height. Word problems apply these formulas to calculate missing values like height, volume, or surface area when other values like radius or volume are given.

Area and circumference of circles

This document defines key terms and formulas related to circles, including circumference and area. It defines a circle as all points equidistant from a given center point. The radius is the distance from the center to the edge, and the diameter runs through the center. Circumference is defined as the distance around the circle and is calculated using either C=2πr or C=πd. Area is calculated as A=πr^2. Several examples are provided to demonstrate calculating circumference and area using these formulas.

Similar Triangles

The document discusses similar triangles and scale factors. It provides examples of similar triangles in nature, art, architecture, and mathematics. It explains the different rules to determine if triangles are similar: AAA (angle-angle-angle), PPP (proportional property), PAP (proportional angles property), and RHS (right-hypotenuse-side). Examples are given applying these rules to prove triangles are similar and calculate missing side lengths or scale factors.

Basic geometrical constructions

Basic geometrical constuctions is how to construct angle by using compass and ruler.
this slide can help students or teachers to construct any angles especially for special angles they are 90 degree, 60 degree, 45 degree and 30 degree.

Surface Area and Volume

This document provides information on calculating surface area and volume for various 3D shapes including prisms, cylinders, pyramids and more. It defines key terms like surface area, volume, and includes formulas and examples for finding the surface area and volume of rectangular and triangular prisms, cylinders, and pyramids using measurements of lengths, widths, heights, radii, etc. Practice problems are provided throughout for additional examples of calculating surface area and volume.

Area of a triangle

To be able to calculate the area of a triangle using trigonometry up to Grade A*. By the end of the lesson, all students will be able to identify when to find the area of a triangle using trigonometry. Most students will be able to calculate the area of a triangle using the formula. Some students will be able to calculate GCSE area of triangle problems using trigonometry up to Grade A*. The document then provides examples and practice problems for students to calculate the area, missing sides and missing angles of triangles using trigonometry.

Introduction to Pythagorean Theorem.pptx

This document discusses triangles and the Pythagorean theorem. It defines different types of triangles, including right triangles. It explains how to calculate the area of triangles. The main focus is on the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Several examples are provided to demonstrate how to use the theorem to solve for missing sides of right triangles.

Square roots

This document provides information about square roots and real numbers. It includes:
1) Examples of finding square roots of perfect squares and estimating other square roots.
2) Classifying different types of real numbers such as rational numbers, irrational numbers, integers, natural numbers, and more.
3) Examples of classifying given numbers as rational, irrational, integers, and other categories.

Arcs and Central Angles

Central angles are angles whose vertex is the center of a circle. A central angle separates a circle into two arcs: a minor arc and a major arc. The measure of a minor arc is equal to the measure of its central angle. The measure of a major arc is equal to 360 degrees minus the measure of the minor arc. The measure of an arc formed by two adjacent arcs is the sum of the measures of the individual arcs. If two minor arcs in the same or congruent circles are congruent, then the corresponding chords are also congruent.

Integers and Operation of Integers

This document provides information and examples about integer operations:
- Addition of integers follows the same rules as normal addition, such as 20 + 10 = 30 and -40 + -60 = -100.
- Subtraction of integers is performed similarly to addition, such as -3 - 7 = -10 and 15 - 9 = 6.
- When multiplying integers, the product is positive if the signs are the same and negative if the signs are different, exemplified as -2 × 6 = -12 and 2 × -3 = -6.
- For division of integers, the quotient is positive if the signs are the same and negative if the signs are different, with examples like 12 ÷ -4 =

Circles - An Introduction

THIS POWERPOINT PRESENTATION ON THE TOPIC CIRCLES PROVIDES A BASIC AND INFORMATIVE LOOK OF THE TOPIC
_________________________________________________
LIKE ...COMMENT AND SHARE THIS PRESENTATION
FOLLOW FOR MORE

Zero and Negative Exponents

- The document discusses how computers use negative exponents to process fractions and percentages during photo processing, which allows apps like Photoshop to shrink photos.
- It explains that negative exponents represent fractions, with the item with the negative exponent moving to the denominator when written as a fraction. This allows computers to perform mathematical operations involving fractions.
- Examples are provided of how negative exponents simplify to fractions through the rule of a-m = 1/am, with the item in the exponent moving between the numerator and denominator.

Estimating square roots

To estimate square roots:
- Recall perfect square numbers like 4, 9, 16, 25, etc.
- The square root of a number is between the greatest smaller perfect square number and the smallest greater perfect square number.
- Examples show estimating the square roots of 32, 129, and 875 by identifying the closest perfect squares above and below.

Perimeter, area and volume

This document discusses perimeter, area, and volume. It begins by defining perimeter as the distance around a shape found by adding all the side lengths. It provides examples of calculating perimeters of rectangles, irregular shapes, and converting between units. It then defines area as a measure of how much surface a shape covers. It gives formulas and examples for finding the areas of rectangles, triangles, parallelograms, trapezoids, and irregular shapes. Finally, it discusses surface area as the total area of all faces of a shape. It provides the surface area formulas and worked examples for cuboids and cubes.

Classifying Angles

This document discusses different types of angles including acute, obtuse, right, and straight angles. It defines an angle as being formed by two rays sharing an endpoint called the vertex. Angles are measured in degrees, with acute angles between 0-90 degrees, obtuse angles between 90-180 degrees, right angles equal to 90 degrees, and a straight angle equaling 180 degrees. It includes examples of each type of angle and encourages identifying them in a game.

Parallelogram area

This document discusses how to calculate the area of parallelograms. It defines key terms like base, height, and parallelogram. The area of a parallelogram is calculated as the base multiplied by the perpendicular height. Examples are provided to demonstrate calculating the area or finding the missing value given the area. The document also briefly discusses rhombuses, noting they are a type of parallelogram with four equal sides and their area is calculated the same way.

11 2 arcs and central angles lesson

This document discusses arcs and central angles in circles. It defines arcs as curved lines formed when two sides of a central angle meet at the center of a circle. There are three types of arcs: minor arcs are inside the central angle and measure less than 180 degrees; major arcs are outside the central angle; and semicircles measure 180 degrees. The measure of an arc depends on its type and the measure of the corresponding central angle. Rules are provided for calculating arc measures using central angles and properties of adjacent arcs. Examples demonstrate finding arc measures using these rules and properties of circles.

Area of trapezium

A trapezium is a quadrilateral with one set of parallel sides. To calculate the area of a trapezium, divide it into a triangle and rectangle by connecting the non-parallel sides. The area formula is (a + b)h/2, where a and b are the parallel sides, h is the height or altitude, and (a - b) gives the base of the triangle formed. An example calculates the area of a given trapezium using this formula.

Polygons

Chapter 3 Polygons
3.1 Definition
3.2 Terminology
3.3 Sum Of Interior Angles Of A Polygon
3.4 Sum Of Exterior Angles Of A Polygon
3.5 Diagonals in one vertex of any Polygon
3.6 Diagonals in any vertices of a Polygon
3.7 Quadrilaterals

Polygons

Polygons

Cones

Cones

Area and circumference of circles

Area and circumference of circles

Similar Triangles

Similar Triangles

Basic geometrical constructions

Basic geometrical constructions

Surface Area and Volume

Surface Area and Volume

Area of a triangle

Area of a triangle

Introduction to Pythagorean Theorem.pptx

Introduction to Pythagorean Theorem.pptx

Square roots

Square roots

Arcs and Central Angles

Arcs and Central Angles

Integers and Operation of Integers

Integers and Operation of Integers

Circles - An Introduction

Circles - An Introduction

Zero and Negative Exponents

Zero and Negative Exponents

Estimating square roots

Estimating square roots

Perimeter, area and volume

Perimeter, area and volume

Classifying Angles

Classifying Angles

Parallelogram area

Parallelogram area

11 2 arcs and central angles lesson

11 2 arcs and central angles lesson

Area of trapezium

Area of trapezium

Polygons

Polygons

surface areas and volume

The document discusses the volume and surface area formulas for different geometric shapes including spheres, cones, cuboids, cubes, and hemispheres. It defines a sphere as a perfectly round object where all points are the same distance from the center, and gives its volume formula as 4/3 * π * r^3 and surface area as 4πr^2. It also provides the surface area formulas for cones, cuboids, cubes, and hemispheres. Examples are given to demonstrate calculating surface areas using the appropriate formulas.

MATHS PRESENTATION

The document thanks the teacher, Mrs. Leena Saji, for giving the opportunity to do a presentation. It also thanks parents and friends for their help in finishing the presentation on time. Finally, it thanks God for everything and bringing the presentation to fruition.

Power point presentation PIYUSH BHANDARI

3-D shapes have length, width, and height. A cube has 6 faces, 12 edges, and 8 vertices. The surface area of a cube can be found by calculating the area of each face, which is the length squared, and multiplying by 6 since there are 6 faces. The surface area of other 3-D shapes like cuboids and cylinders can be found by calculating the total area of all faces, such as using the formula for a cuboid of 2 times the length times width plus 2 times the height times width plus 2 times the length times height. The surface area of a cylinder is 2 times pi times the radius times the height plus 2 times pi times the radius squared.

Maths presentation

The document defines surface area and volume, and provides formulas to calculate the surface areas and volumes of various shapes including cubes, cuboids, cylinders, cones, spheres, and hemispheres. It states that surface area is the total area of the outer surface of an object, while volume is the space occupied by an object. Formulas provided include the lateral and total surface area of cubes, the total surface area and volume of cuboids, the curved and total surface area and volume of cylinders, the curved and total surface area and volumes of cones, and the surface area and volume of spheres and hemispheres.

Circle

Circle is a simple closed shape in Euclidean geometry. It is defined as the set of all points in a plane that are equidistant from a given point, called the center. The distance from the center to any point on the circle is called the radius. A circle can also be defined as a special type of ellipse where the two foci are coincident or as the shape that encloses the maximum area for a given perimeter. Key properties of circles include relationships between circumference, diameter, radius and area. Tangents, chords, and inscribed angles also have important properties related to circles.

Surface areas and volumes

1) The document discusses different geometric shapes including cubes, cuboids, cylinders, cones, and spheres. It provides formulas for calculating the surface area and volume of each shape.
2) Specific examples are given of how to calculate the surface area of a cube, cuboid, cylinder, and cone using various formulas like Surface Area of Cube = 6a^2.
3) The volume formulas for each shape are also outlined, such as the Volume of a Sphere = 4/3 πr^3. Real world examples are given to demonstrate applications of the different shapes.

Digital Text book

This document is about spheres and includes the following key points:
- It defines a sphere as a round object where any cross section would form a circle. The distance from the center to any point on the sphere is the radius.
- It gives the formulas for calculating the surface area and volume of a sphere. The surface area is 4πr^2 and the volume is (4/3)πr^3, where r is the radius.
- It compares a sphere to a cylinder that just contains it, finding that the ratio of their surface areas is 3:2, as is the ratio of their volumes.

Ammu mathe sbed

This document provides definitions and explanations related to spheres. It defines a sphere as a solid generated by revolving a semicircle about its diameter. It lists examples of spheres from daily life like balls and globes. It explains that the diameter of a sphere passes through its center and that the center and radius of the semicircle used to generate the sphere are also the center and radius of the sphere. It defines a hemisphere as half of a sphere divided by a plane passing through the center. It provides formulas for the surface area of a hemisphere and sphere and the volume of a sphere and hemisphere.

Handouts on polygons

The document defines and describes different types of polygons based on the number of sides. It discusses triangles, quadrilaterals, pentagons, hexagons and provides specifics about each type. Regular polygons are defined as those with equal sides and interior angles. Irregular polygons do not have these properties. The document also discusses angles, sides and other geometric properties of polygons.

Mip 2015

Short-cut Formula in Finding the Areas of a Shaded Region of an Inscribed Circle in a Square, and Circumscribing Circle to a Square

MATHS PROJECT

This document discusses the calculation of surface areas and volumes of various 3D shapes. It begins by introducing solids formed by stacking 2D shapes and defines terms like cuboid, cylinder, cone, and sphere. It then provides formulas to calculate the surface area of each shape:
1) The surface area of a cuboid is 2(lb + bh + hl) where l, b, and h are its dimensions.
2) The surface area of a cylinder is 2πrh + 2πr^2, where r is the radius and h is the height.
3) The surface area of a cone is πrl + πr^2, where r is the base radius and l is

surface area and volume

The document provides information on surface area and volume formulas and calculations for basic 3D shapes including prisms, cubes, cylinders, cones, and spheres. It defines key terms like surface area and volume and provides example calculations and formulas for finding the surface area and volume of cubes, rectangular prisms, cylinders, cones, and spheres. Diagrams and examples are included to illustrate the different shapes and how to set up the surface area and volume calculations.

Geom12point6 97

This document provides information about calculating the surface area and volume of spheres. It defines key terms related to spheres such as radius, diameter, center, and hemisphere. It gives the formulas for calculating the surface area (S=4πr^2) and volume (V=4/3πr^3) of spheres. Examples are provided to demonstrate how to use the formulas to find the surface area of a baseball and the radius of a ball bearing given its original volume as a cylinder.

Volume.ppt [recovered]

This document provides information about different 3D shapes - cube, cuboid, cylinder, cone, and sphere. It defines each shape, provides examples, and describes key properties like surface area and volume formulas. Examples are also given to demonstrate calculating the volume of each shape using the relevant formula when given dimensions like side length, radius, or height.

spherical triangles

This document provides information on spherical trigonometry. It defines key terms like great circles, small circles, spherical angles, and spherical triangles. It describes properties of these concepts, such as every great circle passing through the center of a sphere. Formulas for solving spherical triangles are presented, including the Sine Formula, Cosine Formula, and Haversine Formula. Examples show how to use these formulas to calculate unknown sides and angles of spherical triangles given other information.

surface area and volume ppt

1. The document defines various 3D shapes including cubes, cuboids, cylinders, cones, spheres, and hemispheres.
2. It provides the formulas to calculate the surface area and volume of each shape. For cubes, cuboids, cylinders and cones it gives the formulas for total surface area. For spheres and hemispheres it provides the formulas for total surface area, curved surface area, and volume.
3. The document was created collaboratively by several students, with each person responsible for explaining different shapes.

12 6 surface area & volume of spheres

The document discusses the surface area and volume of spheres. It provides formulas for calculating the surface area (S=4πr^2) and volume (V=4/3πr^3) of a sphere. Several examples are worked through, applying these formulas to find surface areas and volumes of spheres given radii and other measurements. The surface area of a baseball is explained to be made up of two congruent shapes resembling two joined circles.

Area and Volume

This document defines and provides formulas for calculating surface areas and volumes of various 3D shapes including cubes, cuboids, cylinders, cones, spheres, and hemispheres. It states that the total surface area of a cube is 6 times the area of one face, and the volume is the cube of the edge length. The total surface area of a cuboid is 2 times the sum of the area of the base and lateral faces, and the volume is length times breadth times height.

Mathematics ppt

A ppt on the surface area and volume of a cone with examples and good definitions which leads to a better understanding........

Digital sherin

The document provides information about circles, including their history, definition, key properties, terminology used to describe parts of a circle, the mathematical constant pi, the relationships between a circle's circumference, diameter, and radius, and formulas for calculating the circumference and area of a circle. It defines a circle as a simple closed curve where all points are equidistant from the center and discusses circles in the contexts of geometry, ancient drawings, the development of mathematics, and everyday use.

surface areas and volume

surface areas and volume

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Power point presentation PIYUSH BHANDARI

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Maths presentation

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Digital Text book

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Ammu mathe sbed

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surface area and volume

surface area and volume

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surface area and volume ppt

surface area and volume ppt

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Mathematics ppt

Digital sherin

Digital sherin

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Diabetes

Diabetes

Polynomials

Polynomials

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Advertising

old age

old age

Chemistry

Chemistry

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Diabetes mellitus

WATER CRISIS “Prediction of 3rd world war”

WATER CRISIS “Prediction of 3rd world war”

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Issue of Shares

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Mansi

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- 2. S P H E R E
- 3. • Circle is the only shape where there is only one edge, no straight lines, and a curve that is completely unified for a full 360 degrees around a single center point. It resolves to One, and thus it is the simplest possible two-dimensional shape. • When we expand this into three dimensions, we can then see that the similar principle applies to the sphere. Physicist Buckminister. Fuller described a sphere as "a multiplicity of discrete events, approximately equidistant in all directions from a nuclear center."
- 4. Some interesting Facts : • It is perfectly symmetrical • It has no edges or vertices (corners) • It is not a polyhedron • All points on the surface are the same distance from the center Surface Area = 4 × π × r2 Volume = (4/3) × π × r3
- 5. Largest Volume for Smallest Surface Of all the shapes, a sphere has the smallest surface area for a volume. Or put another way it can contain the greatest volume for a fixed surface area. Example: if you blow up a balloon it naturally forms a sphere because it is trying to hold as much air as possible with as small a surface as possible. In Nature The sphere appears in nature whenever a surface wants to be as small as possible. Examples include bubbles and water drops, can you think of more?
- 6. The Earth The Planet Earth, our home, is nearly a sphere, except that it is squashed a little at the poles. It is a spheroid, which means it just misses out on being a sphere because it isn't perfect in one direction (in the Earth's case: North-South)
- 7. A sphere can be compressed into a single point, which has no space and no time, and thus exist as the simplest object in the Universe, but the sphere also is the most complex form in the Universe, containing all other things within itself. In a sphere you can draw an infinite number of lines that connect to an infinite number of points (i.e. “events ”) on the surface of the sphere, with all the lines starting from one single center point or nucleus, and all the lines will come out to be the exact same length. This makes the sphere the most complex three- dimensional object that there is; an infinite number of different geometric shapes can be drawn inside of it, by simply connecting different points on the surface of the sphere together.
- 8. Once you stretch or flatten the sphere in any way, you have less symmetry and thus have less flexibility in what can be geometrically created inside. (This may seem hard to understand, but it can be proven mathematically. This also explains why liquid naturally forms into spheres when it is in free-fall and/or in a soap bubble, as the air pressure on the liquid is equal on all sides.) The sphere is also the simplest three-dimensional formation in the Universe for the same reasons as the circle; namely, there is only one edge, perfectly symmetrical in its curvature around a center point, and thus all resolves to One.
- 9. ARYABHATTA–I (476 BC – 550 AD) was 1st to include the computation of areas & volumes of sphere in his literature. ARYABHATTA – I
- 10. Aryabhatta gives a wrong formula for the volume of a sphere. leifj.kkgL;k?kZ fo"dEHkk/kZgreso o`RrQye~A rfUutewysu gra ?kuxksyQy fujo'ks"ke~AA (A. B. Ganitapada. 7) (Half the circumference multiplied by half the diameter is the area of a circle. That multiplied by its own root is the exact volume of a sphere. According to this the volume of sphere = 8 d. 4 d . 4 d 22/322 π = ππ =
- 11. Bhaskara and the later mathematicians give correct formulae for both the surface area and the volume of the sphere. o`r{ks=s ifjf/kxqf.krO;klikn% Qy ;& R{kqi.ka osnS:ifjifjr% dUnqdL;so tkye~A xksyL;Soa rnfi p Qya i`"Bta O;klfu?ue~ "kM~fHkHkZDra Hkofr fu;ra xksyxHksZ ? kuk[;e~AA (Lil. 201) (In a Circle, the circumference multiplied by one- fourth the diameter is the area, which, multiplied by 4, is its surface area going round it like a net round a ball. This (surface area) multiplied by the diameter and divided by 6 is the volume inside the sphere.)
- 12. i.e. area of a circle = = circumference . The surface area of a sphere = 4. area of its great circle = 4πr2 Volume of a sphere = = Surface area . 2 2 r 4 d 4 d π= π = 3 r 3 4 6 r2 π=
- 13. In modern mathematics the formula of volume of sphere can be derived using following method i.e. disc integration to sum the volumes of an infinite no. of circular discs of infinitesimally small thickness stacked centered side by side along the axis from x = 0 where the disc has radius r i.e. y = r to x = r
- 14. At any given x the incremental volume (δv) is given by the product of cross sectional area of the disc at x and its thickness (δx) δv ≈πy2 δx The total volume is summation of all incremental volumes v ≈Σπy2 δx in the limit as δx approaches zero this becomes ∫− π= r r 2 dxyv
- 15. At any given x a right angle triangle connects x, y and r to the origin, hence it follows from the geometry that y2 = r2 – x2 Thus substituting y with a function of x gives This can be now evaluated as ∫− −π= r r 22 dx)xr(v 3 r3 2 r 3 4 3 x xrv π= −π= -r
- 16. Reference :- • Balachandra Rao, S. Indian Mathematics and Astronomy. Jnana Deep Publications. 1994 • Barathi Krishna Tirtha. Vedic Mathematics. Motilal Banrsidas. 1992 • Bhanu Murthy. T.S. A Modern Introduction to Ancient Indian Mathematics. Wiley Eastern. 1992 4. Durant, Will. The Story of Civilization. Part I: Our Oriental Heritage. Simon and Schuster, New York, 1954.