What is the curved surface area of a cone?
Step-by-step solution
Idea: Cut the cone along a slant edge and unroll it: it becomes a sector of a circle of radius l whose curved edge (arc) is the base circle, length 2πr. Cut that sector into very thin triangles of height l and add their areas.
- Unroll the curved surface: it is a sector of a circle of radius l (the slant height). Its arc s goes round the base, so s = 2πr.1 mark
- Cut the sector into many thin pieces; each is almost a triangle with base b and height l, area 12bl. Adding: CSA = 12(b1 + b2 + … )l = 12 × (arc length) × l.1 mark
- So CSA = 12 × 2πr × l = πrl. (The same comes from the sector formula: area = πl2 × θ360° with θ360° = rl.)½ mark
- Adding the circular base: TSA = πrl + πr2 = πr(l + r). If only the height is known, find l from l2 = h2 + r2.½ mark
Check: Sector check: a sector of radius l and angle θ has area πl2 × θ360°. Its arc 2πl × θ360° must equal 2πr, so θ360° = rl and area = πl2 × rl = πrl ✓.
Answer to write in the exam
Curved surface unrolled = sector of radius l, arc = 2πr
Area of sector = 12 × arc × radius = 12 × 2πr × l
∴ CSA of cone = πrl; TSA = πr(l + r)
Common mistakes that cost marks
- Using the height h instead of the slant height l in πrl.
- Writing 13πr2h (the volume) for the surface area.
- Adding πr2 when only the curved surface is asked for (e.g. a joker’s cap or a tent has no base).
How this can come in the exam
The curved surface area of a cone of radius 3 cm and height 4 cm is
- 12π cm2
- 15π cm2
- 24π cm2
- 9π cm2
Show answer
(B) 15π cm2
l = √(9 + 16) = 5; CSA = π × 3 × 5 = 15π cm2.
A sector of radius 10 cm and angle 216° is folded into a cone. Find the radius of its base and its curved surface area in terms of π.
Show answer
rl = 216360 = 35 ⇒ r = 6 cm. CSA = π × 6 × 10 = 60π cm2 (= π × 102 × 35 ✓).Try one yourself
Find the total surface area of a cone with radius 5 cm and slant height 12 cm, in terms of π.
Show answer
π × 5 × (12 + 5) = 85π cm2.
More questions like this
- What is the volume of a cone?
- Here is another hands-on activity that uses modelling clay (earlier this used to be called ‘plasticine’). Carefully mold the clay into a solid cylinder and measure its radius and height. Then reshape the same portion of clay into identical cones, each having the same radius and the same height as the cylinder. You will find that you are able to make exactly three such cones.
- A right triangle ABC with sides 5 cm, 12 cm and 13 cm is rotated through 360° about the side with length 12 cm. Find the volume of the solid so obtained.
- Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
- Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.