Lesson 10 8 volume
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1. The document is a lesson on calculating volume for rectangular prisms and triangular prisms. It provides the formulas for volume (V=l×w×h for rectangles, V=B×h for triangles where B is the area of the base) and examples of using the formulas to solve problems. 2. Students are asked to practice calculating volumes of different shapes using the given formulas and by finding the area of various bases for triangular prisms. 3. The key aspects covered are defining volume, the volume formulas, and examples of applying the formulas to solve volume problems for different shapes.
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Lessons 12 13 merged
This document contains a math lesson on calculating the volume of rectangular prisms. It provides examples of three rectangular prisms with different heights but the same length and width, and has students write expressions for the volume of each. It then has students recognize that these expressions all represent the area of the base multiplied by the height. Students are asked to determine the volumes of additional prisms using this area of base times height formula.
C0211014019
1. The document presents a new formula for calculating the area of an isosceles triangle derived by the author through applying Pythagorean theorem.
2. The formula is: Area of isosceles triangle = b × (4a^2 - b^2)/4, where a is the length of two equal sides and b is the length of the unequal side (base).
3. The author provides examples and step-by-step working to demonstrate the application of the new formula, as well as verification by comparing the results to existing Heron's formula.
G6 m5-c-lesson 12-s
This document provides examples and problems about calculating the volume of rectangular prisms. It begins by showing different rectangular prisms and having students write expressions for the volume of each using length, width, and height. It explains that the volume can also be written as the area of the base times the height. Students then practice calculating volumes using both methods. Later problems involve calculating volume when given the area of the base and height or vice versa. The goal is for students to understand that the volume of a rectangular prism is the area of its base multiplied by its height.
Area of a trapezoid
This document discusses properties of polygons and their areas. It provides formulas to calculate the areas of trapezoids, rhombuses, and similar polygons. It also outlines principles of equal areas, such as parallelograms and triangles having equal areas if they have congruent bases and altitudes. Several problems are solved as examples, such as finding missing lengths or areas of trapezoids, rhombuses, and proving polygons have equal areas.
Area of a trapezoid
This document provides examples of calculating the area of a trapezoid using the formula Area = 1⁄2(b+B)h, where b and B are the two bases and h is the height. It demonstrates finding the height when it is not explicitly given by using properties of isosceles triangles or the Pythagorean theorem. In one example, it shows dividing the larger base into pieces to find the height of two congruent triangles formed. In another, it notes that if the base angles are 45 degrees, the triangles formed will be isosceles and the shared height can be used for the trapezoid area.
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2) Identifying local extrema, maxima and minima, of a function using first derivatives.
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Invention of the plane geometrical formulae - Part II
Invention of the plane geometrical formulae - Part IIInternational Journal of Engineering Inventions www.ijeijournal.com
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
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1. The author invented a new formula for calculating the area of an isosceles triangle based on Pythagorean theorem.
2. The formula is: Area of an isosceles triangle = b × 4a2 - b2/4, where b is the base and a is the length of the two equal sides.
3. The author provides two examples applying the new formula and verifies the results match when using Heron's formula. This proves the new formula is valid for calculating the area of an isosceles triangle.
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This document contains a math lesson on calculating the volume of rectangular prisms. It provides examples of three rectangular prisms with different heights but the same length and width, and has students write expressions for the volume of each. It then has students recognize that these expressions all represent the area of the base multiplied by the height. Students are asked to determine the volumes of additional prisms using this area of base times height formula.
C0211014019
1. The document presents a new formula for calculating the area of an isosceles triangle derived by the author through applying Pythagorean theorem.
2. The formula is: Area of isosceles triangle = b × (4a^2 - b^2)/4, where a is the length of two equal sides and b is the length of the unequal side (base).
3. The author provides examples and step-by-step working to demonstrate the application of the new formula, as well as verification by comparing the results to existing Heron's formula.
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This document provides examples and problems about calculating the volume of rectangular prisms. It begins by showing different rectangular prisms and having students write expressions for the volume of each using length, width, and height. It explains that the volume can also be written as the area of the base times the height. Students then practice calculating volumes using both methods. Later problems involve calculating volume when given the area of the base and height or vice versa. The goal is for students to understand that the volume of a rectangular prism is the area of its base multiplied by its height.
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This document discusses properties of polygons and their areas. It provides formulas to calculate the areas of trapezoids, rhombuses, and similar polygons. It also outlines principles of equal areas, such as parallelograms and triangles having equal areas if they have congruent bases and altitudes. Several problems are solved as examples, such as finding missing lengths or areas of trapezoids, rhombuses, and proving polygons have equal areas.
Area of a trapezoid
This document provides examples of calculating the area of a trapezoid using the formula Area = 1⁄2(b+B)h, where b and B are the two bases and h is the height. It demonstrates finding the height when it is not explicitly given by using properties of isosceles triangles or the Pythagorean theorem. In one example, it shows dividing the larger base into pieces to find the height of two congruent triangles formed. In another, it notes that if the base angles are 45 degrees, the triangles formed will be isosceles and the shared height can be used for the trapezoid area.
Chapter 4 review
This document covers key concepts in calculus including:
1) Approximating changes in a function using derivatives, finding local linearizations.
2) Identifying local extrema, maxima and minima, of a function using first derivatives.
3) Defining critical points as points where the derivative is zero or undefined.
4) Relating properties of the first and second derivatives to characteristics of the original function like concavity, inflection points, and local extrema.
Invention of the plane geometrical formulae - Part II
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International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.Invention of the plane geometrical formulae - Part II
1. The author invented a new formula for calculating the area of an isosceles triangle based on Pythagorean theorem.
2. The formula is: Area of an isosceles triangle = b × 4a2 - b2/4, where b is the base and a is the length of the two equal sides.
3. The author provides two examples applying the new formula and verifies the results match when using Heron's formula. This proves the new formula is valid for calculating the area of an isosceles triangle.
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1) This lesson teaches students the formulas for calculating the volume of right rectangular prisms and cubes. It provides examples of using the formulas to find the volume when given the length, width, height or area of the base.
2) Students complete exercises that explore how changes to the lengths or heights affect the volume. They discover that if the height is doubled, the volume is also doubled, and if the height is tripled the volume is tripled.
3) No matter the shape, when the side lengths are changed by the same fractional amount, the volume changes by that fractional amount cubed. For example, if the sides are halved, the volume is one-eighth of the original.
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How to get the area of a trapezoid? Well, all you need to do is to browse this presentation to get ideas out of it.
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The document discusses methods for calculating the area of a parallelogram. It explains that a parallelogram can be divided into a rectangle or two triangles to calculate its area using the formula of base times height. It provides an example of using the formula to find the area of a parallelogram when given the base and height measurements.
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2. The formula is: Area = b(4a^2 - b^2)/4, where b is the base and a is the length of the two equal sides.
3. The author provides two examples calculating the areas of isosceles triangles using the new formula and verifies the results using Heron's formula.
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2. The volume of a prism can be calculated using the formula V = Bh, where B is the area of the base and h is the height of the prism.
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2) Equate coefficients of like terms
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4) Substitute the values to express the expression as a sum of partial fractions.
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Rectangular and triangular prisms have formulas to calculate their volume (V).
For a rectangular prism, V = (bh)h, where b is the width, h is the height, and the product bh gives the area of the rectangular base.
For a triangular prism, V = (1/2)bh, where b is the base, h is the height. The area of the triangular base is calculated as (1/2)bh.
Both formulas multiply the area of the base by the height to determine the volume of the entire prism.
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G6 m5-c-lesson 13-t
1) This lesson teaches students the formulas for calculating the volume of right rectangular prisms and cubes. It provides examples of using the formulas to find the volume when given the length, width, height or area of the base.
2) Students complete exercises that explore how changes to the lengths or heights affect the volume. They discover that if the height is doubled, the volume is also doubled, and if the height is tripled the volume is tripled.
3) No matter the shape, when the side lengths are changed by the same fractional amount, the volume changes by that fractional amount cubed. For example, if the sides are halved, the volume is one-eighth of the original.
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How to get the area of a trapezoid? Well, all you need to do is to browse this presentation to get ideas out of it.
Area parallelograms
The document discusses methods for calculating the area of a parallelogram. It explains that a parallelogram can be divided into a rectangle or two triangles to calculate its area using the formula of base times height. It provides an example of using the formula to find the area of a parallelogram when given the base and height measurements.
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F0261036040
1. The author invented a new formula for calculating the area of an isosceles triangle based on Pythagorean theorem.
2. The formula is: Area = b(4a^2 - b^2)/4, where b is the base and a is the length of the two equal sides.
3. The author provides two examples calculating the areas of isosceles triangles using the new formula and verifies the results using Heron's formula.
Triple product of vectors
The document discusses the triple product of vectors. The scalar triple product of vectors A, B, and C equals A dot (B cross C) and represents the volume of the parallelepiped formed by the three vectors. The vector triple product of vectors A, B, C, and D equals A cross (B cross C) and represents a vector perpendicular to the plane formed by vectors B and C.
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This document discusses how to calculate the volumes of pyramids and cones. It provides the formulas that the volume of a pyramid is 1/3 base area multiplied by height, and the volume of a cone is 1/3 pi r^2 h. Examples are given of applying these formulas to find the volumes of different pyramids and cones. It also mentions that cross-sections of three-dimensional shapes can be analyzed and gives examples of describing cross-sections as a hexagon or triangle.
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This document provides a tutorial on solving one-variable equations with fractions through 13 examples of different types of fractional equations. The examples cover equations of the form a/bx = c, -a/bx = c, a/bx = -c, a/bx = cx + d, a/bx = c/dx + e, and other variations involving addition, subtraction, and combinations of fractions and variables.
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The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant base. It provides examples and discusses key characteristics of exponential graphs such as their domains, ranges, and asymptotic behavior. The document also covers transformations of exponential graphs and using exponential functions to model compound interest over time.
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This document provides examples and exercises about calculating the volumes of cubes and rectangular prisms using formulas. It begins with examples of calculating the volume of a cube with sides of 2 1/4 cm and a rectangular prism with a base area of 7/12 ft^2 and height of 1/3 ft. The exercises then involve calculating volumes of cubes and prisms when dimensions are changed, identifying relationships between dimensions and volumes, and writing expressions for volumes.
Parallel lines
Parallel lines have the same slope. To write the equation of a parallel line, first find the slope (m) of the original line by rewriting its equation in y=mx+b form. Then use the same m value and a new given point (x,y) to write the parallel line's equation in y=mx+b form by solving for b. For example, to find the equation of a line parallel to 2x-3y=17 through the point (9, -3), first get the original line in y=mx+b form to find m, then use m and (9, -3) to write the parallel line's equation.
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The document discusses calculating the volumes of pyramids and cones. It provides the formulas that the volume of a pyramid is one-third the area of the base times the height, and the volume of a cone is one-third pi times the area of the base times the height. It then gives examples of calculating the volumes of different pyramids and cones using the formulas and given measurements of base and height.
Volume of Prisms
1. The document discusses how to calculate the volume of prisms. Volume is the number of cubic units needed to fill a space, and a cubic unit is the space occupied by a cube of unit measure.
2. The volume of a prism can be calculated using the formula V = Bh, where B is the area of the base and h is the height of the prism.
3. Examples are provided to demonstrate calculating the volume of rectangular and triangular prisms using measurements of the base and height.
Partial Fractions Linear Term To A Power
1) Expressions with (ax + b)k in the denominator can be decomposed into terms of the form: ax, b, (ax + b)2, (ax + b)3, etc.
2) To express 8x/(x-3)3 as partial fractions, express it as individual fractions with the factors on the denominator and then with a common denominator.
3) Substitute values for x to solve for the coefficients A and B.
Partial Fractions Quadratic Term
This document discusses decomposing expressions with quadratic terms in the denominator into partial fractions. It provides an example of expressing (x^2-4)(x-4) as partial fractions. The steps shown are:
1) Express the expression with a common denominator of (x^2-4)(x-4)
2) Equate coefficients of like terms
3) Solve the system of equations to find the values of A, B, and C
4) Substitute the values to express the expression as a sum of partial fractions.
Partial Fractions Linear Term
This document discusses expressing a rational expression with linear terms in the denominator as a sum of partial fractions. It shows working through an example, factorizing the denominator into linear factors and then setting up equations to solve for the coefficients of the partial fractions by either comparing coefficients or direct substitution. The example is solved to give the partial fraction decomposition as 10/(x-6) - 2/(x-3).
Geometry unit 10.4
This document discusses the effects of changing dimensions on the perimeter and area of geometric figures. It provides examples of how multiplying or doubling individual dimensions, such as the height of a rectangle or radius of a circle, affects the perimeter and area. It also examines how proportionally changing multiple dimensions, such as doubling both the base and height of a rectangle, impacts perimeter and area. The key points are that individually changing a dimension proportionally changes the perimeter or area, while proportionally changing multiple dimensions results in a similar figure. The document aims to build understanding of these relationships through worked examples and practice problems.
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Similar to Lesson 10 8 volume
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Rectangular and triangular prisms have formulas to calculate their volume (V).
For a rectangular prism, V = (bh)h, where b is the width, h is the height, and the product bh gives the area of the rectangular base.
For a triangular prism, V = (1/2)bh, where b is the base, h is the height. The area of the triangular base is calculated as (1/2)bh.
Both formulas multiply the area of the base by the height to determine the volume of the entire prism.
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Three-Dimensional Geometry discusses spatial relations and three-dimensional figures. It explains that three-dimensional figures have faces, edges, and vertices. The document provides examples and formulas for calculating the volumes of prisms, cylinders, cones, pyramids and cubes. It also discusses surface area and provides examples and formulas for calculating surface areas of prisms and cylinders.
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This document provides information about calculating the area of different shapes. It defines area as the amount of surface a figure covers. It provides the formulas for calculating the area of rectangles, parallelograms, and triangles. It then works through examples of finding the area of various shapes by plugging the measurements into the appropriate formula. This includes calculating the area of a parallelogram-shaped stained glass window and the area of a rectangular yard.
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The document discusses volume formulas for different 3D shapes. It provides the formulas to calculate the volume of a rectangular prism, cube, cylinder, cone, and rectangular pyramid. Examples are given to demonstrate how to use the formulas to find the volume of each shape by plugging in the given measurements for length, width, height, radius, etc.
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The document provides instructions for calculating the surface area of various 3D shapes. It defines surface area as the sum of the areas of all surfaces of a solid figure. It then works through examples of finding the surface area of a triangular prism and rectangular prism by calculating the area of each face and adding them together. Students are provided practice problems to find the surface area of additional prisms.
Lesson 10 7 surface area
The document provides instructions for calculating the surface area of various 3D shapes. It defines surface area as the sum of the areas of all surfaces of a solid figure. It then works through examples of finding the surface area of a triangular prism and rectangular prism by calculating the area of each face and adding them together. Students are provided practice problems to find the surface area of additional prisms.
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See also:
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016 ;
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial ;
http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra ;
http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems ;
http://www.slideshare.net/JamesSmith245/solution-of-the-llp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-using-reflections-and-rotations ;
http://www.slideshare.net/JamesSmith245/simplied-solutions-of-the-clp-and-ccp-limiting-cases-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/additional-solutions-of-the-limiting-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/an-additional-brief-solution-of-the-cpp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-ga .
Similar to Lesson 10 8 volume (8)
Solution Strategies for Equations that Arise in Geometric (Cliff ord) Algebra
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Final exam review sheet # 2 2015
This document contains a review sheet for a math final exam. It includes two parts - a multiple choice section with 37 questions covering various math concepts, and a short answer section with 7 word problems requiring calculations and explanations. The review sheet provides the questions, space to write answers, and an answer key in the back to check work.
Final exam review sheet # 3 2015
This document contains a multi-part math exam review with multiple choice and short answer questions. It provides practice problems covering topics like geometry, ratios, equations, expressions, and word problems. The review is designed to help students prepare for their math final exam.
Final exam review sheet # 1 2015
This document contains a review sheet for a math final exam. It includes multiple choice and short answer questions covering topics like geometry, algebra, ratios, and word problems. It also provides the answers to the multiple choice section. The short answer questions require showing work and include problems finding areas, writing equations, comparing ratios, and solving word problems involving money.
Mod 5 lesson 12 13
This document contains notes from a math lesson on volume. It discusses determining the volume of composite figures using decomposition into simpler shapes. Students will practice finding the volume of various objects. The document contains examples of area problems and notes for students to solve.
G6 m5-c-lesson 12-t
This lesson teaches students about calculating the volume of rectangular prisms using two different formulas: 1) length × width × height and 2) area of the base × height. Students work through examples calculating the volume of various rectangular prisms using both formulas. They learn that it does not matter which face is used as the base, as the volume will be the same. The lesson reinforces that volume can be expressed in multiple equivalent ways and emphasizes using the area of the base times the height.
Mod 5 lesson 9
1) The document outlines a math lesson plan for a week in May that includes topics on polygons, area, surface area, and volume.
2) On Tuesday, students will work on problem sets for Lesson 9 and 13, which cover finding the perimeter and area of polygons on the coordinate plane.
3) On Thursday, students will work on a Lesson 15 worksheet, and on Friday they are asked to bring in a rectangular prism from home to create a net and label edge lengths.
G6 m5-b-lesson 9-t
Lesson 9 focuses on determining the area and perimeter of polygons on the coordinate plane. Students will find the perimeter of irregular figures by using coordinates to find the length of sides joining points with the same x- or y-coordinate. Students will also find the area enclosed by a polygon by composing or decomposing it into polygons with known area formulas. The lesson provides examples of calculating perimeter and area, as well as exercises for students to practice these skills by decomposing polygons in different ways.
G6 m5-b-lesson 9-s
This lesson teaches students how to determine the area and perimeter of polygons on a coordinate plane. It includes examples of calculating area and perimeter of polygons. Students are given exercises to calculate the area of various polygons, determine both the area and perimeter of shapes, and write expressions to represent the area calculated in different ways. The lesson aims to help students practice finding area and perimeter of polygons located on a coordinate plane.
Mod 5 lesson 8
This document discusses a lesson on drawing polygons on the coordinate plane. The lesson objectives are for students to use absolute value to determine distances between integers on the coordinate plane in order to find side lengths of polygons. The document includes examples of polygons drawn on the coordinate plane and questions about determining their areas and shapes. It closes by asking students to complete an exit ticket to assess their understanding of determining areas of polygons using different methods, and how the polygon shape influences the area calculation method.
G6 m5-b-lesson 8-t
This document provides a lesson on drawing polygons on the coordinate plane. It includes 4 examples of plotting points and drawing polygons to connect those points. It then provides the name of each polygon drawn and how to calculate its area, whether by using formulas for basic shapes like triangles or decomposing complex shapes into simpler ones. The document emphasizes using coordinates to determine side lengths and plotting points accurately on the coordinate plane in order to find polygon areas.
G6 m5-b-lesson 8-s
This document provides examples and exercises for plotting polygons on a coordinate plane and calculating their areas. It includes examples of plotting triangles, rectangles, and other polygons given their vertices and calculating their areas. It then gives practice problems for students to plot polygons on their own, name the shapes, determine the areas using formulas, and write expressions to represent the area calculations. The final problems involve finding missing vertices of polygons when given some vertices and the total area.
G6 m5-b-lesson 8-note sheet
1. The document contains a student worksheet with multiple geometry problems. It asks the student to plot points, identify shapes, and calculate areas. The first problem involves plotting points A, B, and C, identifying the triangle shape, and calculating its area.
2. The second problem has the student plot points E, F, and G, name the triangle shape, and calculate its area.
3. The third problem involves plotting points K, L, M, and N, naming the quadrilateral shape, and calculating its total area.
G6 m5-b-lesson 7-t
This document describes a lesson on determining distance on the coordinate plane. Students will use absolute value to find the lengths of line segments between integer coordinates. They will recognize when segments are vertical or horizontal based on shared x- or y-coordinates. For vertical or horizontal segments, distance is calculated by subtracting or adding the absolute values of the different coordinates. Students practice finding distances and representing their work in a table with a column showing the calculation.
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- 1. Name ________________________________ Date __________________ Mrs. Labuski / Mrs. Rooney Period _____ Lesson 10-8 Volume VOCABULARY DEFINITION EXAMPLE The number of cubic units needed to fill a space. VOLUME Formula = (of rectangular length x width x height prism) V=lxwxh V=3x2x4 Answer is always written V = 24 units3 in units3 or cubic units. V = Bh VOLUME notice the capital B! (of triangular B = area of the base See back of notes page prism) Answer is always written in units3 or cubic units Now you try: Find the area of the main shape in front V=lxwxh V=lxwxh (base) and then multiply V=4x2x3 V = 12 x 10 x 8 that by the V=8x3 V = 120 x 8 height of the main figure V = 24 V = 960 V = 24 ft3 V = 960 yd3
- 2. To find the volume of a triangular prism V = Bh (think of capital B as A – Area) First find the area of the base. A=½x5x4 A = ½ x 20 A = 10 (= B) Then multiply the result by the height of the prism. V = Bh V = 10 x 3 V = 30 un3 V = Bh V = Bh (B=Area of bases:) (B=Area of bases:) A=½x4x2 A=½x3x2 A=½x8 A=½x6 A = 4 (= B) A=3 V=4x4 V=3x5 V = 16 un3 V = 15 un3
- 3. Name ________________________________ Date __________________ Mrs. Labuski / Mrs. Rooney Period _____ Lesson 10-8 Volume VOCABULARY DEFINITION EXAMPLE VOLUME (of rectangular prism) VOLUME (of triangular prism) Now you try: V=lxwxh V=lxwxh V = ___________ V = ___________ V = ___________ V = ___________ V = ___________ V = ___________ V = ___________ V = ___________
- 4. To find the volume of a triangular prism V = Bh First find the area of the base. A = _________ A = _________ Then multiply the result by the height of the prism. V = Bh V = ___________ V = ___________ V = Bh V = Bh (B=Area of bases:) (B=Area of bases:) A = ½(bh) A = ½(bh) A = _________ A = _________ A = _________ A = _________ V = __________ V = __________ V = __________ V = __________