Chapter 8 covers concepts related to solid geometry, including definitions and properties of three-dimensional shapes such as prisms, cuboids, cubes, cylinders, pyramids, cones, and spheres. It details methods for calculating surface area and volume for each shape, providing examples and exercises for practice. The chapter emphasizes the distinction between two-dimensional and three-dimensional objects and introduces key geometric principles.

1. Chapter 8
Surface Area
AndVolume

2. Chapter 8 Surface Area AndVolume
• 8.1 Introduction to Solid Geometry
• 8.2 Prism
• 8.3The Cuboid AndThe Cube
• 8.4 Cylinders
• 8.5 Pyramids
• 8.6 Right Circular Cone
• 8.7 Sphere

3. 8.1 Introduction to Solid Geometry
• All the geometric shapes discussed previously till now i.e. polygons,
circles etc. are planar shapes. They are called two dimensional
shapes i.e. generally speaking they have only length and width. In
the real world however every object has length, width and height.
Therefore they are called three dimensional objects. No single
plane can contain such objects in totality.
• Consider the simplest example of a three dimensional shape - a
brick.

4. • Shown in Figure 8.1 is a brick with length 10 cm, breadth 5 cm and
height 4 cm.There cannot be a single plane which can contain the
brick.
• A brick has six surfaces and eight vertices. Each surface has an area
which can be calculated.The sum of the areas of all the six surfaces
is called the surface area of the brick.
• Apart from surface area a brick has another measurable property.
i.e. the space it occupies.This space occupied by the brick is called
its volume.
8.1 Introduction to Solid Geometry

5. 8.1 Introduction to Solid Geometry
• Every three dimensional (3-D) object occupies a finite volume.The 3
-D objects or geometric solids dealt within this chapter are
• (1) Prism
(2) Cube
(3) Right circular cylinder
(4) Pyramid
(5) Right circular cones and
(6) Sphere
• Apart from defining these objects, methods to calculate their
surface area and volume are also incorporated in this chapter.

6. 8.2 Prism
• Any solid formed by joining the corresponding vertices of two
congruent polygons is called a prism.
• The two congruent polygons are called the two bases of the prism.
The lines connecting the corresponding vertices are called lateral
edges and they are parallel to each other. The parallelogram
formed by the lateral edges are called lateral faces.
• (See figure 8.2)Prisms are of two types depending on the angle
made by the lateral edges with the base.
Figure 8.2

7. 8.2 Prism
• If the lateral edges are perpendicular to the base the prism is called
a right prism. If the lateral edges are not perpendicular to the base
the prism is called an oblique prism.
• Consider the right prism shown in Figure 8.3
• ABCDEF is a prism DABC & DFED are congruent and seg. AF, seg.
CD and seg. BE are perpendicular to the planes containing DABC
& DFED.
Figure 8.3
Also in DABC m  ABC = 90o , (seg. AB) = 3 cm
& (seg. AC) = 5 cm

8. 8.2 Prism
• The surface area of the prism is the sum of the surface areas of all
its surfaces.
• base is unknown and height is 3 cm.
Figure 8.3
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐴 ∆𝐴𝐵𝐶 + 𝐴 ∆𝐹𝐸𝐷 + 𝐴 𝑞𝑢𝑎𝑑 𝐴𝐵𝐸𝐹 +
𝐴 𝑞𝑢𝑎𝑑 𝐴𝐶𝐷𝐹 + 𝐴(𝑞𝑢𝑎𝑑 𝐵𝐶𝐷𝐸)
𝐴 ∆𝐴𝐵𝐶 = 𝐴 ∆𝐹𝐸𝐷 =
1
2
𝑏𝑎𝑠𝑒 ∙ ℎ𝑒𝑖𝑔ℎ𝑡

9. 8.3The cuboid and the cube
• A book, a match box, a brick are all examples of a cuboid.
• The definition of a cuboid is derived from that of the prism.
• A cuboid is a solid formed by joining the corresponding vertices of
two congruent rectangles such that the lateral edges are
perpendicular to the planes containing the congruent rectangles.
Figure 8.4 shows a cuboid.
Figure 8.4
w

10. 8.3The cuboid and the cube
• As can be seen in Figure 8.4 the cuboid has six surfaces and each one is
congruent and parallel to the one opposite to it.
• Thus there is a pair of three rectangles which goes to make a cube.
• It is already known that the area of a rectangle is the product of its
length and width, l x w = Area.
• The surface area of a cuboid is the sum of the areas of all its faces,
• The volume of a cuboid is the number of unit cubes that are required to
fill the cuboid completely.
Figure 8.4
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝐶𝑢𝑏𝑜𝑖𝑑 = 2 ℓ ∙ 𝑤 + 𝑤 ∙ ℎ + ℓ ∙ ℎ
w
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝐶𝑢𝑏𝑜𝑖𝑑 = 𝑙 ∙ 𝑤 ∙ ℎ

11. Cube
• Now consider a cuboid made of 6 congruent squares instead of a
pair of three rectangles. This structure has l = w = h, i.e. the length,
width and height are the same. Such a cuboid is called a cube.
• A cube is a square right prism with the lateral edges of the same
length as that of a side of the base. (See figure 8.5)
figure 8.5

12. Cube
• As can be seen in Figure 8.5 in a cube the length, width and height
are equal. Therefore the surface of a cube is essentially six squares
of equal length.
• The area of a square is the square of its length. The total surface
area of the cube is the sum of the areas of all its 6 surfaces. If length
of one side of the cube is l the area of one surface is ℓ2 , therefore
The surface area of the cube = 6 ℓ2
.
figure 8.5
TheVolume of the cube = ℓ3.

13. Exercises
Example 1
Give the surface area and the
volume of a given cuboid at figure
8.6.
figure 8.6

14. Exercises
• Example 2
• Draw a right hexagonal prism with height 5 cm and length of one
side of the hexagon = 2.5 cm.
• Solution:

15. Exercises
• Example 3
• When is a cuboid a cube ?
• Example 4
• What is the difference between a right prism and an oblique prism ?
• Example 5
• What is the altitude of a prism ?
• Example 6
• If the volume of a cube is 27. Find the surface area.
• Example 7
• If the height of a cuboid is zero it becomes a (a) prism (b) cube (c)
rectangle.

16. 8.4 Circular Cylinders
• Pillars, pipes etc. are examples of circular cylinders encountered
daily.
• A circular cylinder is two circles with the same radius at a finite
distance from each other with their circumferences joined.
• The definition of a circular cylinder is a prism with circular bases.
The line joining the centers of the two circles is called the axis.
• If the axis is perpendicular to the circles it is a right circular cylinder
otherwise it is an oblique circular cylinder
• The lateral area of a right circular cylinder is the product of the
circumference and the vertical distance between the two circles or
the altitude ‘h’
figure 8.7

17. 8.4 Circular Cylinders
where ’r’ is the radius
of the circle.
• The total area must include apart from the lateral area, the areas of
the two circles.
• The volume of a right circular cylinder,
figure 8.7
Lateral Area = circumference ∙ ℎ
Lateral Area = 2𝜋𝑟ℎ
Area of the circle = 𝜋𝑟2 Total area = 2𝜋ℎ + 2𝜋𝑟2
or = 2𝜋r(h + r)
Volume = Area of the circle ∙ h Volume = 𝜋𝑟2h

18. 8.5 Pyramids
• A pyramid is a polygon with all the vertices joined to a point outside
the plane of the polygon.
• If the polygon is regular then the pyramid is called a regular
pyramid and is named by the polygon which forms its base.
• If the base is a square the pyramid is called a regular square
pyramid, if it is a pentagon the pyramid is called a regular
pentagonal pyramid and so on and so forth.
• A regular pyramid has a property called slant height which is the
perpendicular distance between the vertex and any side of the
polygon.

19. 8.5 Pyramids
• The lateral area of a regular pyramid is defined using this
parameter.
• Where ‘p’ is the perimeter and ‘ℓ’ is the slant height.
Where BA = Base Area
figure 8.8
Lateral Area of a regular Pyramid (LA) =
1
2
𝑝ℓ
Perimeter (P) =(seg.AB) + (seg.BC) + (seg.CA)
Total Area = (LA + BA)
Volume of a pyramid =
1
3
𝐵𝐴 ℎ

20. 8.5 Pyramids
• Example
• For a regular square pyramid if the length of a base = 4 and the
height = 6. Find the volume.
• Solution:
since length of the base = 4
Base area = BA=(4)2=16
Volume = 1/3 (BA)h
= (1/3) (16) (6) = 32 cubic units

21. 8.6 Right circular cone
• A right circular cone is a circle with all points on its circumference
joined to a point equidistant from all of them and outside the plane
of the circle.
• The lateral area of the right circular cone
=
1
2
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ∙ 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 =
1
2
(2𝜋𝑟) ℓ
• The total areaTA = (LA + BA ), where BA = base area = 𝜋𝑟2
LA = 𝜋𝑟 ℓ
TA = 𝜋𝑟 ℓ + 𝜋𝑟2
= 𝜋𝑟(ℓ + r) Volume =
1
3
𝐵𝐴 ℎ =
1
3
𝜋𝑟2
h
figure 8.9

22. 8.7 Sphere
• The simplest example of a sphere is a ball. One can call here that a
circle is a set of points in a plane that are equidistant from one point
in the plane. If this is extended to the third dimension, we have all
points in space equidistant from one particular point forming a
sphere.
• The sphere has only one surfaces and its Surface Area = 4𝜋𝑟2
The volume of a sphere =
4𝜋𝑟3
3
figure 8.10

23. Exercises
• Example 1: If the lateral area of a right circular cylinder is 24p and
its radius is 2 .What is its height.
• Example 2: Find the total area of the right circular cylinder with a
radius of 10 units and a height of 5 units.
• Example 3: What is the radius of the right circular cylinder with
volume 18p cubic units and height = 2.
• Example 4: If the perimeter and slant height of a regular pyramid
are 10 and 3 respectively. Find its lateral area.
• Example 5: If the volume of a sphere is 36p. Find its radius and
surface area.

24.  end 
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