VOLUME

VOUMES OF THREE – DIMENSIONAL FIGURES

PRISMS
Recall that a prism is a solid with a constant cross-section.
If the area of cross –section is A and the height is h, then the volume is
V=Ah

- The cuboid and the cylinder are examples of prisms. The volume of a cuboid is given by
V = lbh (The cross –section is a circle)
- The volume of a cylinder is given by
V= πr h (The cross –section is a circle)
- For both of these solids, the volume is given by
V= Ah, where A is the area of cross-section.
Example
The cross-section of a right angled prism is a triangle for which the shorter are 5cm and 6cm . The prism is 20cm long . Find its volume
Solution:

- Now multiply by the height
Volume = 20×15
= 300
∴ There volume is 300cm3


EXERCISE 3.5.A
1. 1. The cross-section of a prism is a regular hexagen with side 0.8m. The prime is 1.6 long. Find the volume of the prism.
2. 2. The cross-section of a prism is a right angled triangle with sides 12cm, 9cm and 15cm, and with height 25cm. Find the volume of the prism.
PYRAMIDS AND CONES

- The pyramid tapers to a point from a base, which is usually a rectangle.
- The cone tapers to a point from a base, which is a circle.
- For both these solids, the volume is a third that of the corresponding prism with the same base. The volume of a pyramid is a third of the corresponding cuboid

∴ The volume of a cone is a third that of the corresponding cylinder

Example
A pyramid has a square base. Its height is 7cm and its volume is 56cm3. Find the side of the base.
Solution
- Suppose the side of the base is x2 cm2
- Formula for the volume of a pyramid

∴ The side of the base is 4.9cm

EXERCISE 3.5.B
1. 1. A pyramid has a square base of side 10cm and volume 500cm3. Find its height.
2. 2. A cone has height 12cm and volume 50cm3. Find its base radius.

SPHERES
If a sphere has radius r, then its volume is

Example 1:
A sphere has radius 8.7cm. Find its volume.
Solution
Apply the formula

∴ The volume is 2760cm3
Example 2.
A sphere has volume 100cm3. Find its radius
Solution

∴ The radius is 2.88m

EXERCISE 3.5C
1. 1. A sphere has volume 1.6m3
a) Find its radius.
b) Find its surface area.
2. 2. A sphere has surface area 56cm2
a) Find its radius
b) Find its volume
SOLUTIONS
EXERCISE 3.1.A
1. a) EFGH b) HDCG c) AD d)EA
2. a) GCDH and GCBF b)FG,FE and FB
EXERCISE 3.1 B
1. a) A Match box b) A die
2. a)Cuboid b) Sphere c) Cylinder d)Cone
EXERCISE 3.2A
1. 1. a)


b)




2. 2. a) A pyramid
b) A prism
EXERCISE 3.3A
1. a) ABCD and BCGF

b) AB and DC

c) EC and AG

d) EB and FC

EXERCISE 3.3.B

∴ The angle between the line AG and the face ABCD is 35.30
2. 2. Given
AB=20M, AD=30m and VA =VB=VC=VD=25m



∴ The angle between V A and ABCD is 43.90
EXERCISE 3.3C
Given
- A square base of side 20m
- VA = =VC=VD =15M
- Angle between the planes VAB and ABCD = ?

∴ The angle between the planes VAB and ABCD is 48.190
2. Given
- Length of the prism 8cm
- Side of the triangle 5cm
- Angle between the planes ADEB and BEFC =?

∴ The angle between the planes ADEB and BEFC is 600
EXERCISE 3.4A
1. Given
1. A square base of side 0.8m and height 1.1m
Formula = Base area = (0.8×0.8)×2
= 0.64 ×2
= 1.28
Then other areas
= (0.8 ×1.1) ×4=3.52
- The total area = 1.28 + 3.52=4.8
∴ The surface area of the crate 4.8m2
2. Given
- A room with length = 5.2m, height=2.5m and width= 4.5m
-The area of the wall = length × width
= 5.2 × 4.5
= 23.4 m2
- The area of the wall =(height × length) ×2 + (height ×width)×2
= (2.5×5.2)x2 + (2.5×4.5 )×2
= 26+22.5
= 48.5m2
- The total area = 23.4m2 + 48.5m
= 71.9m2
∴ The surface are of the wall and the wiling is 71.9m2
EXERCISE 3.4B
1. 1. Given
- Height = 2cm
- Radius = 4cm
- Formula = 2 πr (r+h)
= 2 π×4(4+2)
= 8π×6
= 48π
=150.72
∴The surface area is 151cm2
2. Given
Length =15cm
Radius=3cm
Area of the wall = length × width

Area of the wall = 2πr ×L
=2×π×3×15
= 90 π
= 282.6cm2
EXERCISE 3.4C
1. a)surface area of the prisms=
- Base area = Rectangle=L×W
-The area of rectangle are
=(21×40) + (29×40)
=840 ×1,160 +800
=2,800cm2

But the prism have 2 triangles
210×2 = 420cm2
-The total area is 2,8000+420=3,220
The surface area of the prism is 3,220cm2
Given
Side = 8cm
Length=30cm
-A regular pentagon has 5 sides
-Area of a regular pentagon has 5 rectangles which are equal.
- The area of rectangles are
Formula = L×n
= (30×8)×5
=240×5
=1,200cm2
Base area

-

- For base areas, that is top and bottom area there are 10 triangles.
- Therefore 22.0224×10=220.224 = 220
Hence
The total area = 1,200 + 220 = 1,420cm2
∴ The surface area of the prism is 1420cm2
EXERCISE 3.4 D
Given
-A square base of side = 10cm and height is 12cm. the base area =5×5=50
=10×10
=1002

- Total are of all triangles 65cm2 × 4 = 260cm2
∴ The surface area of triangular faces is 260cm2