Find the surface area of a composite 3-D shape
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The dimensions of a cube and a cuboid, which is length (l), breadth (b) and height (h).
height
height
breadth
length
breadth
length
-
The surface area of a three- dimensional composite shape is the total area of the surfaces
exposed.
-
To find the surface area of a composite 3-D shape, use the formulae for finding the area of a
rectangle or a square.
Surface area = Length × Breadth
-
The units of surface area area the same as the units for area, cm2 and m2.
EXAMPLE 1
Find the surface area of the composite shape.
1 cm
1 cm
1 cm
1 cm
2 cm
Solution:
1 cm
1 cm
A
B
1 cm
1 cm
2 cm
Surface area of cube A = 5 × (1 cm × 1 cm)
= 5 cm2
Surface area of cuboid B = [3 × (1 cm × 1 cm)] + [3 × (2 cm × 1 cm)]
= 9 cm2
Total surface area = (5 + 9) cm2
= 14 cm2
EXAMPLE 2
The diagram shows a cube R and a cuboid S.
6 cm
3 cm
R
S
10 cm
Calculate the total surface area of the composite solid.
Solution:
Surface area of cube R = [5 × (6 cm × 6 cm)] + (3 cm × 6 cm)
= 180 cm2 + 18 cm2
= 198 cm2
Surface area of cuboid S = [2 × (10 cm × 6 cm)] + [2 × (10 cm × 3 cm)] + (3 cm × 6 cm)
= 120 cm2 + 60 cm2 + 18 cm2
= 198 cm2
Total surface area = (198 + 198) cm2
= 396 cm2
EXAMPLE 3
The diagram shows a composite solid consisting of three identical cuboids.
2 cm
5 cm
8 cm
Find the total surface area of the solid.
Solution:
2 cm
5 cm
8 cm
4 cm
8 cm P
Q
Surface area of cuboid P
= [2 × (8 cm × 2 cm)] + [2 × (5 cm × 2 cm)] + (8 cm × 5 cm) + (5 cm × 4 cm)
= 32 cm2 + 20 cm2 + 40 cm2 + 20 cm2
= 112 cm2
Surface area of cuboid Q
= [2 × (8 cm × 4 cm)] + [2 × (8 cm × 5 cm)] + (5 cm × 4 cm)
= 64 cm2 + 80 cm2 + 20 cm2
= 164 cm2
Total surface area = (112 + 164) cm2
= 276 cm2
EXAMPLE 4
4 cm
(a)
8 cm
P
12 cm
Q
10 cm
Find the total surface area of the shaded parts.
Solution:
Surface area of P = 4 cm × 4 cm
= 16 cm2
Surface area of Q = 10 cm × 12 cm
= 120 cm2
Total surface area = (16 + 120) cm2
= 136 cm2
(b)
The diagram below shows a composite 3-D shape. Shapes A and C are identical.
A
C
B
4 cm 3 cm
3 cm
Find the total area surface of the shaded parts.
Solution:
Surface area of A = surface area of C
= 2 × (4 cm × 3 cm)
= 2 × 12 cm2
= 24 cm2
Surface area of B = 3 cm × 3 cm
= 9 cm2
Total surface area = (24 + 9) cm2
= 33 cm2
(c)
Calculate the surface area of the shaded region.
5 cm
A
B
8 cm
C
Solution:
Surface area of A + B = 5 cm × 10 cm
= 50 cm2
Surface area of C = 18 cm × 5 cm
= 90 cm2
Total surface area = (50 + 90) cm2
= 140 cm2
(d)
6 cm
A
2 cm
B
C
7 cm
Solution:
Surface area of A = 6 cm × 2 cm
= 12 cm2
Surface area of B = 2 cm × 2 cm
= 4 cm2
Surface area of C = 7 cm × 2 cm
= 14 cm2
Total surface area = (12 + 4 + 14) cm2
= 30 cm2
(e)
4 cm
P
3 cm
Q
8 cm
R
2 cm
Solution:
Surface area of P = 4 cm × 4 cm
= 16 cm2
Surface area of Q = 7 cm × 4 cm
= 28 cm2
Surface area of R = 2 cm × 2 cm
= 4 cm2
Total surface area = (16 + 28 + 4) cm2
= 48 cm2
(f)
2 cm
P
Q
2 cm
3 cm
7 cm
R
8 cm
10 cm
Solution:
Surface area of P = 2 cm × 2 cm
= 4 cm2
Surface area of Q = 7 cm × 2 cm
= 14 cm2
Surface area of R = 10 cm × 8 cm
= 80 cm2
Total surface area = (4 + 14 + 80) cm2
= 98 cm2