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发展经济学1 
MENSURATION 
PRESENTED BY 
1. ARJUN RASTOGI
What is Mensuration 
 Mensuration in its literal meaning is 
to measure, Its generally used where 
geometrical figures are concerned, 
Where one has to determine various 
physical quantities such as area , 
volume , length measuring these 
quantities is called mensuration also 
it is used where quantities like 
speed, velocity and acceleration etc 
are concerned
2D SHAPES 
• A shape that only has two dimensions 
(such as width and height) and no 
thickness. Squares, Circles, Triangles, etc 
are two dimensional objects. 
发展经济学3
SOME OF SHAPES IN MENSURATION 
2D 
1) Square 
 2) Rectangle 
 3) Parallelogram 
 4) Rhombus 
5) Triangle 
6) Trapezoid 
7) Circle
a square is a regular quadrilateral, which 
means that it has four equal sides and four 
equal angles (90-degree angles, or right 
angles). It can also be defined as 
a rectangle in which two adjacent sides 
have equal length. A square 
with vertices ABCD would be 
denoted ABCD.
PERIMETER 
• The Perimeter is 
• 4 times the side length 
Perimeter = 4a 
• Perimeter is the distance 
around a closed figure and is 
typically measured in 
millimeters (mm), centimeters 
(cm), meters (m) and 
kilometers (km). These units 
are related as follows: 
• 10 mm = 1 cm 
• 100 cm = 1 m 
• 1000 m = 1 km 
发展经济学6
AREA 
 AREA OF SQUARE IS
 A rectangle is any quadrilateral with four right 
angles. It can also be defined as an equiangular 
quadrilateral, since equiangular means that all of 
its angles are equal (360°/4 = 90°).. A rectangle 
with four sides of equal length is a square. The 
term oblong is occasionally used to refer to a 
non-square rectangle A rectangle 
with vertices ABCD would be denoted 
as ABCD.
PERIMETER 
The Perimeter is 
2 times the (width +height ) 
Perimeter = 2(w+h) 
W = width 
H = height
AREA 
The Area is 
the width times 
the height 
Area = w × h 
Width =w 
Height = h
A parallelogram is a (non self-intersecting) 
quadrilateral with two pairs 
of parallel sides. The opposite or facing 
sides of a parallelogram are of equal 
length and the opposite angles of a 
parallelogram are of equal measure
12 
PERIMETER 
• The Perimeter is 
2 times the (base + 
side length): 
• Perimeter = 2(b+s)
AREA OF PARALLELOGRAM 
• The Area is the 
• base times the height: 
• Area = b × h 
• (h is at right angles to b) 
发展经济学13
a rhombus , is a simple (non-self-intersecting) 
quadrilateral all of whose four 
sides have the same length. Another name 
is equilateral quadrilateral, since 
equilateral means that all of its sides are 
equal in length. The rhombus is often 
called a diamond
PERIMETER 
The Perimeter is 
4 times "s" (the side 
length) 
because all sides are 
equal in length: 
Perimeter = 4s
AREA OF RHOMBUS 
Area of rhombus 
the altitude times the side 
divided by 2 
Area = (p × q)/2 
Rhombus = side length 
of rhombus 
h = height of rhombus 
d1 = long diagonal of rhombus 
d2 = short diagonal of 
rhombus
 A triangle is a polygon with 
three edges and three vertices. It is one 
of the basic shapes in geometry. A 
triangle with vertices A, B, and C is 
denoted 
 TYPES OF TRIANGLE 
1. ISOSCLES 
2. EQUILATERAL 
3. SCALENE
• 
PERIMETER 
•Perimeter of triangle 
Sum of sides 
3s 
As perimeter is sum of 
sides which can be also 
written as 
P=A+B+C
AREA OF TRIANGLE 
A triangle is half the area of a 
rectangle. To find the area of a 
triangle, you use the rectangle 
formula (base times height) 
and divide it in half. 
A = base • height 
2
A convex quadrilateral with at least one 
pair of parallel sides is referred to as 
a trapezoid . The parallel sides are called 
the bases of the trapezoid and the other 
two sides are called the legs or the lateral 
sides (if they are not parallel; otherwise 
there are two pairs of bases).
Perimeter of trapezoid 
the sum of all side lengths: 
Perimeter = a+b+c+d
Area of trapezoid 
• The area of a polygon is the number of square 
units inside that polygon. Area is 2-dimensional 
like a carpet or an area rug. A trapezoid is a 4- 
sided figure with one pair of parallel sides. For 
example, in the diagram to the right, the bases 
are parallel. To find the area of a trapezoid, take 
the sum of its bases, multiply the sum by the 
height of the trapezoid, and then divide the 
result by 2, The formula for the area of a 
trapezoid is 
发展经济学22
A circle is a simple shape Of that is the 
set of all points in a plane that are at a 
given distance from a given point, 
the centre. The distance between any of 
the points and the centre is called 
the radius. It can also be defined as the 
locus of a point equidistant from a fixed 
point.
The perimeter is of circle 
2 times radius into pi 
Circumference = 2 × π × r
AREA OF CIRCLE 
The Area is the 
side length squared: 
Area = a2 = a × a 
The distance around a circle is called 
its circumference. The distance across a 
circle through its center is called 
its diameter. We use the Greek letter 
The area of a circle is the number of square 
units inside that circle
INTRODUCTION 
 TOTAL SURFACE AREA(TSA) 
Total surface are is the sum of the areas of all the sides 
of Three dimensional figure 
 LATERAL SURFACE AREA (LSA) 
Lateral area refers to the surface area of a 3D object 
such as prism, cylinder, cone etc., Lateral surface 
area is the sum of all sides of a 3D object except its 
top and bottom bases. 
 VOLUME 
The amount of 3-dimensional space an object 
occupies. Capacity.
3D SHAPES 
An object that has height, width and depth, 
like any object in the real world.
SOME OF SHAPES IN 
MENSURATION 
3D 
 1) Cube 
2) Rectangular 
Prism (Cuboids) 
 3) Cylinder 
4) Cone 
5) Sphere and 
Hemisphere 
6) Prism 
7) Pyramid
A cuboid is a box-shaped solid object. It 
has six flat sides and all angles are right 
angles. 
And all of its faces are rectangles.
 Lateral surface area of the 
cuboid 
= perimeter of rectangular 
base * height 
= 2(l + w)h square units 
= 2h(l + w) square units 
Total surface area of the 
cuboid 
= lateral surface area + area of 
base + area of top 
= [2h(l + w) + lw + lw] square 
units 
= (2hl + 2hw + 2lw) square 
units 
= [2(lh + wh + lw)] square units
 a cube is a three-dimensional solid object 
bounded by six square faces, facets or 
sides, with three meeting at each vertex.
THE TSA OF CUBE 
IS =6A2 
The LSA OF CUBE 
IS 
=4A2 
VOLUME OF CUBE 
 =A 3
 A cone is a three-dimensional geometric 
shape that tapers smoothly from a flat 
base (frequently, though not necessarily, 
circular) to a point called the apex or 
vertex.
Total surface area = rlπ + πr² 
 lateral surface area of a cone = πrl 
Volume of a cone
 cylinder is one of the most basic 
curvilinear geometric shapes, the surface 
formed by the points at a fixed distance 
from a given line segment, the axis of the 
cylinder.
The total surface area (TSA) of 
a cylinder with radius r and 
height h is 
 LSA OF CYLINDER=2πrh 
Volume OF CYLINDER= 
πr2h
PYRAMID 
A pyramid is a structure whose outer 
surfaces are triangular and converge to a 
single point at the top, making the shape 
roughly a pyramid in the geometric sense.
TOTAL SURFACE AREA = 
LATERAL SURFACE 
AREA= 
VOLUME=Area of the 
base * Height * 1/3
a solid geometric figure whose two ends 
are similar, equal, and parallel rectilinear 
figures, and whose sides are 
parallelograms.
PRISM 
TOTAL SURFACE =LSA + 
2B (lateral surface area x 
area of the base) 
Lateral surface area =p x 
h (perimeter of the base x 
the height) 
VOLUME=B x h (area of 
the base x height)
A sphere is a perfectly round geometrical 
and circular object in three-dimensional 
space that resembles the shape of a 
completely round ball.
SPHERE 
TOTAL SURFACE AREA=4πr2 
VOLUME=4/3πr3
Mensuration

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Mensuration

  • 2. What is Mensuration  Mensuration in its literal meaning is to measure, Its generally used where geometrical figures are concerned, Where one has to determine various physical quantities such as area , volume , length measuring these quantities is called mensuration also it is used where quantities like speed, velocity and acceleration etc are concerned
  • 3. 2D SHAPES • A shape that only has two dimensions (such as width and height) and no thickness. Squares, Circles, Triangles, etc are two dimensional objects. 发展经济学3
  • 4. SOME OF SHAPES IN MENSURATION 2D 1) Square  2) Rectangle  3) Parallelogram  4) Rhombus 5) Triangle 6) Trapezoid 7) Circle
  • 5. a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.
  • 6. PERIMETER • The Perimeter is • 4 times the side length Perimeter = 4a • Perimeter is the distance around a closed figure and is typically measured in millimeters (mm), centimeters (cm), meters (m) and kilometers (km). These units are related as follows: • 10 mm = 1 cm • 100 cm = 1 m • 1000 m = 1 km 发展经济学6
  • 7. AREA  AREA OF SQUARE IS
  • 8.  A rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°).. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle A rectangle with vertices ABCD would be denoted as ABCD.
  • 9. PERIMETER The Perimeter is 2 times the (width +height ) Perimeter = 2(w+h) W = width H = height
  • 10. AREA The Area is the width times the height Area = w × h Width =w Height = h
  • 11. A parallelogram is a (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure
  • 12. 12 PERIMETER • The Perimeter is 2 times the (base + side length): • Perimeter = 2(b+s)
  • 13. AREA OF PARALLELOGRAM • The Area is the • base times the height: • Area = b × h • (h is at right angles to b) 发展经济学13
  • 14. a rhombus , is a simple (non-self-intersecting) quadrilateral all of whose four sides have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond
  • 15. PERIMETER The Perimeter is 4 times "s" (the side length) because all sides are equal in length: Perimeter = 4s
  • 16. AREA OF RHOMBUS Area of rhombus the altitude times the side divided by 2 Area = (p × q)/2 Rhombus = side length of rhombus h = height of rhombus d1 = long diagonal of rhombus d2 = short diagonal of rhombus
  • 17.  A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted  TYPES OF TRIANGLE 1. ISOSCLES 2. EQUILATERAL 3. SCALENE
  • 18. • PERIMETER •Perimeter of triangle Sum of sides 3s As perimeter is sum of sides which can be also written as P=A+B+C
  • 19. AREA OF TRIANGLE A triangle is half the area of a rectangle. To find the area of a triangle, you use the rectangle formula (base times height) and divide it in half. A = base • height 2
  • 20. A convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid . The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides (if they are not parallel; otherwise there are two pairs of bases).
  • 21. Perimeter of trapezoid the sum of all side lengths: Perimeter = a+b+c+d
  • 22. Area of trapezoid • The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional like a carpet or an area rug. A trapezoid is a 4- sided figure with one pair of parallel sides. For example, in the diagram to the right, the bases are parallel. To find the area of a trapezoid, take the sum of its bases, multiply the sum by the height of the trapezoid, and then divide the result by 2, The formula for the area of a trapezoid is 发展经济学22
  • 23. A circle is a simple shape Of that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.
  • 24. The perimeter is of circle 2 times radius into pi Circumference = 2 × π × r
  • 25. AREA OF CIRCLE The Area is the side length squared: Area = a2 = a × a The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter. We use the Greek letter The area of a circle is the number of square units inside that circle
  • 26.
  • 27. INTRODUCTION  TOTAL SURFACE AREA(TSA) Total surface are is the sum of the areas of all the sides of Three dimensional figure  LATERAL SURFACE AREA (LSA) Lateral area refers to the surface area of a 3D object such as prism, cylinder, cone etc., Lateral surface area is the sum of all sides of a 3D object except its top and bottom bases.  VOLUME The amount of 3-dimensional space an object occupies. Capacity.
  • 28. 3D SHAPES An object that has height, width and depth, like any object in the real world.
  • 29. SOME OF SHAPES IN MENSURATION 3D  1) Cube 2) Rectangular Prism (Cuboids)  3) Cylinder 4) Cone 5) Sphere and Hemisphere 6) Prism 7) Pyramid
  • 30. A cuboid is a box-shaped solid object. It has six flat sides and all angles are right angles. And all of its faces are rectangles.
  • 31.  Lateral surface area of the cuboid = perimeter of rectangular base * height = 2(l + w)h square units = 2h(l + w) square units Total surface area of the cuboid = lateral surface area + area of base + area of top = [2h(l + w) + lw + lw] square units = (2hl + 2hw + 2lw) square units = [2(lh + wh + lw)] square units
  • 32.  a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
  • 33. THE TSA OF CUBE IS =6A2 The LSA OF CUBE IS =4A2 VOLUME OF CUBE  =A 3
  • 34.  A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
  • 35. Total surface area = rlπ + πr²  lateral surface area of a cone = πrl Volume of a cone
  • 36.  cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder.
  • 37. The total surface area (TSA) of a cylinder with radius r and height h is  LSA OF CYLINDER=2πrh Volume OF CYLINDER= πr2h
  • 38. PYRAMID A pyramid is a structure whose outer surfaces are triangular and converge to a single point at the top, making the shape roughly a pyramid in the geometric sense.
  • 39. TOTAL SURFACE AREA = LATERAL SURFACE AREA= VOLUME=Area of the base * Height * 1/3
  • 40. a solid geometric figure whose two ends are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.
  • 41. PRISM TOTAL SURFACE =LSA + 2B (lateral surface area x area of the base) Lateral surface area =p x h (perimeter of the base x the height) VOLUME=B x h (area of the base x height)
  • 42. A sphere is a perfectly round geometrical and circular object in three-dimensional space that resembles the shape of a completely round ball.
  • 43. SPHERE TOTAL SURFACE AREA=4πr2 VOLUME=4/3πr3