The height of a cone is 16 cm, and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone.
Answer: l = √(162 + 122) = 20 cm. CSA = π × 12 × 20 = 240π ≈ 753.6 cm2; TSA = π × 12 × 32 = 384π ≈ 1205.76 cm2 (π = 3.14).
Step-by-step solution
Given: h = 16 cm; r = 12 cm
To find: CSA and TSA
To find: CSA and TSA
Idea: The formulas need the slant height, so first use l2 = h2 + r2. Here 12 : 16 : 20 is 3 : 4 : 5.
- l = √(h2 + r2) = √(256 + 144) = √400 = 20 cm.1 mark
- CSA = πrl = 3.14 × 12 × 20 = 3.14 × 240 = 753.6 cm2.1 mark
- TSA = πr(l + r) = 3.14 × 12 × 32 = 3.14 × 384 = 1205.76 cm2.1 mark
CSA = 240π ≈ 753.6 cm²; TSA = 384π ≈ 1205.76 cm² (π = 3.14). With π = 22/7: ≈ 754.29 cm² and ≈ 1206.86 cm².
Check: TSA − CSA = 1205.76 − 753.6 = 452.16 = 3.14 × 144 = area of the base ✓.
Answer to write in the exam
l = √(162 + 122) = √400 = 20 cm
CSA = πrl = 3.14 × 12 × 20 = 753.6 cm2
TSA = πr(l + r) = 3.14 × 12 × 32
∴ CSA = 753.6 cm2, TSA = 1205.76 cm2
Common mistakes that cost marks
- Using the height 16 cm in πrl instead of the slant height 20 cm.
- Writing l = 16 + 12 = 28 instead of using Pythagoras.
How this can come in the exam
MCQ (1 mark)
The slant height of a cone of height 8 cm and radius 6 cm is
- 14 cm
- 10 cm
- √28 cm
- 48 cm
Show answer
(B) 10 cm
√(64 + 36) = √100 = 10 cm.
Short answer (2 marks)
A cone has height 24 cm and radius 7 cm. Find its total surface area. (π = 227)
Show answer
l = √(576 + 49) = 25 cm. TSA = 227 × 7 × (25 + 7) = 22 × 32 = 704 cm2.Try one yourself
Find the curved surface area of a cone of height 9 cm and radius 12 cm, in terms of π.
Show answer
l = 15 cm; CSA = π × 12 × 15 = 180π cm2.
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