SURFACE AREA OF THREE-DIMENSIONAL FIGURES
Consider the diagram which shows a cube of 5cm . It has 6 faces, which are squares of side 5cm. So the total surface area is
2. Consider the diagram which show a cuboid which is a cm by b cm by c cm. It has 6 faces which are rectangles: two are a cm by b cm, two are b cm by c cm and two are c cm by a cm. so its total surface area is
=(a cm × b cm)×2+ (b cm × c cm) x 2 + (a cm× c cm)×2
=2ab cm2 + 2bc cm2+2ac cm2
=(2ab+2bc+2ac)cm2
Example 1:
Find the surface area of a cuboid which is 12cm by 10cm by 8cm.
Solution:
Given:
a = 12cm, b = 10cm and c = 8cm
Formula = (2ab+2bc)cm2
=(2×12×10)+(2×10×8)+(2×12×8)
= 240 + 160 = 192
= 592
The surface area is 592cm2
EXERCISE 3.4A
1. 1. A tea crate has a square base of side 0.8m, and its height is 1.1m. Find the surface area of the crate.
2. 2. A room is 5.2m long, 2.5m high and 4.5m wide. Find the surface area of the wall and the wiling.
CYLINDER
The diagram show a cylinder, which has height him and top radius r cm.
- - The surface of a cylinder consists of a circular top and bottom, and curved side.
- -Imagine cutting the cylinder down the side, and unfolding it. The curved side becomes a rectangle, with height h and width the circumference of the cylinder.
Example
Find the curved surface area of a cylinder which is 6cm high and with radius 4cm.
EXERCISE 3.4 B
1. A tin of shoe polish is 2cm high and 4cm in radius. Find its surface area.
2. A paint roller is a cylinder which is 15cm long and with radius 3cm. Find the area of wall it can cover in one revolution
PRISMS
- Recall that a prism is a solid with a constant cross section
- In many cases the cross-section is a triangle.
- The surface of a prism consists of the two cross-sections and the sides. In particular, the surface of a triangular and the sides. In particular, the surface of a triangular prism consists of two triangles and three rectangles.
Example
A prism has a cross-section which is a triangle with sides 5cm, 12cm and and 13cm. Its length is 10cm. Find Its surface area
Solution.
-The surface consists of:
· . Two triangles of side 5,12and 13
· . Three rectangles: 5, by 10, 12 by 10 and 13 by 10
Note:
The two triangles are right angled.
- The area of each triangle is
The areas of the rectangle are: Formula=l ×w
=(50×10)+ (12×10) =(13×10)
=50 +120+130
=300cm2
-Total area =(2×30)+300
=60+300
=360cm2
The surface area is 360cm2
EXERCISE 3.4C
Find the surface area of the prisms shown below
2.The cross-section of a prism is a regular pentagon of side 8cm. The prism is 30cm long. Find the surface area of the prism.
PYRAMIDS
The surface area of a pyramid is the sum of the base area and the area of the triangular sides. Sometimes you need to use Pythagoras’ theorem to find the area of the sides. The following example shows the method.
Example
A pyramid has a square base of side 10cm and height 15cm.
- Take a line from the vertex V to the middle of one of the sides of the base. The vertical rise of this line is 15cm.
- The horizontal run of this line is
- Hence the length of the line is
Exercise 3.4 D
1. A pyramid has a square base of side 10cm and height 12cm. Find the surface area of its triangular faces.
2. A pyramid has a rectangular base which is 40cm by 60cm. Its vertex is 20cm above the centre of the base. Find the total surface area of the pyramid.
CONES
-The height h, of a cone is its perpendicular height. It is not the length of the slanting edge.
-The length of the slanting edge, l is given by Pythagoras’ theorem.
-Imagine cutting along the side of the cone unfolding. You would get a sector of a circle. The radius of this circle is l and the arc length is the circumference of the cone base, 2πr. Hence the area of the curved side, which is the area of this sector, is the area of the circle, πl2
π L2 Is reduce in the ratio of the circumference of the cone base and the complete circle
∴ The total area of the cone is
πr2 + πrl = πr(r+L)
Example
A cone has base radius 4cm and height 3cm . Find its curved surface area.
Solution
The slant height l is l is given by
EXERCISE 3.4E
Find the surface area of these cones.
a) a) With base radius 5cm and height 12cm.
b) b) With base radius 7m and height24m
2.The base radius of a cone is c cm and its surface area is 160 cm2. Find its slant height and hence find its height.
SPHERES
-A sphere is a round solid, like a ball.
If the sphere has radius r, then its surface area A is
A = 4π2
Example
Find the surface area of sphere of radius 0.46m
Solution
Given,
Radius r = 0.46
Area = 4π2
= 4× π(46)2
= 2.660
∴ The surface area is 2.66m2
EXERCISE 3.4F
1. Find the radio of the spheres with area
a) a) 64πcm b) 0.44m2
2. A sphere has surface area 48cm2. Find its radius
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