The surface area of the cone is πr² + πrl. The πr² part is the round base. The πrl part is the curved side. Here l is the slant height, the length down the sloping side.
Open a cone flat and you get a circle and a fan.
S = πr² + πrl
Hello, curious mind!
What is the surface area of a cone?
It is the total area of the outside: the round base plus the curved side.
| Part | Formula | What it covers |
|---|---|---|
| Base | πr² | The flat circle at the bottom |
| Curved surface | πrl | The sloping side only |
| Total surface | πr² + πrl = πr(r + l) | Base and side together |
Why the curved part is πrl
Cut the cone down one side and lay it flat. The side opens into a sector: a fan cut from a big circle of radius l. That big circle would have area πl².
The fan’s curved edge was the rim of the base, so it is 2πr long. The big circle’s edge is 2πl. So the fan is 2πr ÷ 2πl = r ÷ l of the big circle.
Fan area = (r ÷ l) × πl² = πrl
The fan angle is (r ÷ l) × 360°. A cone with r = 4 and l = 9 opens into a 160° fan.
Remember this
Area uses the slant height l. Volume uses the height h.
How to find the surface area of the cone: 3 worked examples
Example 1: r and l are given
A cone has a base radius of 4 cm and a slant height of 9 cm. Find its total surface area.
Curved part: π × 4 × 9 = 36π.
Base: π × 4² = 16π.
Total = 52π ≈ 163.4 cm².
Example 2: r and h are given (find l first)
A cone has a radius of 20 cm and a height of 21 cm, measured straight up the middle.
h, r and l make a right-angled triangle, with l the longest side. By Pythagoras’ theorem: l = √(20² + 21²) = √841 = 29 cm.
Curved part: π × 20 × 29 = 580π. Base: π × 20² = 400π.
Total = 980π ≈ 3,078.8 cm².
Example 3: total area is given (find r)
A closed cone has a total surface area of 33π cm² and a slant height of 8 cm. Find r.
πr² + 8πr = 33π. Divide by π: r² + 8r − 33 = 0.
Factorise: (r + 11)(r − 3) = 0. See quadratic equations.
A length cannot be negative, so r = 3 cm. Check: 9π + 24π = 33π.
Open cone or closed cone?
Ask one question: is there a flat circle to cover? If not, use the curved part only.
| Object | Base? | Use |
|---|---|---|
| Party hat, ice-cream cone, funnel | No | πrl |
| Solid cone, cone with a lid | Yes | πr² + πrl |
| Tent with a groundsheet | Yes (sheet) | πrl + πr² |
A party hat (made-up numbers)
A card hat has r = 8 cm and l = 20 cm. Card = π × 8 × 20 = 160π ≈ 502.7 cm². No base: a head goes there.
A desert camp tent (made-up numbers)
A cone tent has a radius of 2 m and a height of 2.1 m.
l = √(2² + 2.1²) = √8.41 = 2.9 m.
Canvas for the side: π × 2 × 2.9 = 5.8π ≈ 18.22 m².
Add a groundsheet of 4π: total 9.8π ≈ 30.79 m².
Common surface area of a cone mistakes
- Using h in place of l. In Example 2, π × 20 × 21 = 420π. The true curved area is 580π.
- Adding a base the object does not have. The party hat becomes 160π + 64π = 224π.
- Taking the width across as r. A cone 8 cm across with l = 9 cm has r = 4 cm. Using 8 gives 72π for the side, twice the true 36π.
- Squaring wrongly when finding l. Writing 20² as 40 gives √(40 + 441) ≈ 21.9, not 29.
- Leaving the unit as cm. A surface is an area, so the unit is squared.
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- Uses the height h in place of the slant height l: CNC, concept not clear.
- Adds the base for an open cone: DRC, did not read carefully.
- Puts the diameter where the radius goes: CWV, copied a wrong value.
- Squares wrongly in l = √(r² + h²): CMT, calculation mistake.
- Leaves the unit as cm: BPT, bad presentation.
The Free BMI Maths Test finds which of the 6 mistake types your child makes, and their real maths level. About the Free BMI Maths Test
Test prep corner
Cones on the SAT: what the sheet gives
College Board puts “Area and volume” in SAT Geometry and Trigonometry. Its digital SAT practice tests have a reference sheet with the cone volume, V = ⅓πr²h. The sheet has no surface area formula for any solid. So you must know πrl yourself. We wrote both questions in the SAT style; they are not past test items.
SAT-style: A right circular cone has a base radius of 12 and a height of 35. Its lateral (curved) surface area is kπ. Find k.
l = √(12² + 35²) = √1,369 = 37. Lateral area = π × 12 × 37 = 444π, so k = 444. Trap: 12 × 35 = 420 uses the height, not the slant.
SAT-style: A cone has a slant height of 26 and a lateral surface area of 260π. What is its volume? (A) 260π (B) 800π (C) 866⅔π (D) 2,400π
πr × 26 = 260π, so r = 10. Then h = √(26² − 10²) = 24. V = ⅓ × π × 100 × 24 = 800π, answer (B). Traps: (D) drops the ⅓; (C) uses l = 26 as the height.
Next, test yourself on the SAT-style maths practice questions page.
Facts checked on 12 November 2026 on satsuite.collegeboard.org.
Quick check: 5 cone surface area questions
1. r = 3 cm and l = 7 cm. What is the curved surface area?
Yes! π × 3 × 7 = 21π.
Not quite. 30π adds the base. The question asks for the curved part.
Not quite. There is no 2 in πrl.
2. A closed cone has r = 2 cm and l = 6 cm. What is its total surface area?
Not quite. Add the base, 4π.
Yes! 4π + 12π = 16π.
Not quite. πrl is 12π, not 24π.
3. A cone has r = 9 cm and h = 40 cm. What is the slant height?
Not quite. Square both, add, then take the root.
Not quite. Do not subtract: l is the longest side.
Yes! 81 + 1,600 = 1,681 = 41².
4. Which formula gives the card needed for an ice-cream cone (no lid)?
Not quite. There is no base to cover.
Yes! Open cone: curved part only.
Not quite. That is the volume.
5. A cone has r = 5 cm and l = 15 cm. What angle is the fan when you open it flat?
Yes! (5 ÷ 15) × 360° = 120°.
Not quite. r ÷ l is one third, and a third of 360° is 120°.
Not quite. That would need r ÷ l = ½.
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Questions people ask about the surface area of a cone
What is the formula for the surface area of the cone?
Total = πr² + πrl, with r the base radius and l the slant height. The curved part alone is πrl.
How do I find the slant height?
Use l = √(r² + h²). For r = 20 cm and h = 21 cm, l = √841 = 29 cm.
What is the difference between curved and total surface area?
Curved area is the sloping side only, πrl. Total area adds the base circle, πr².
How do I find the area of an open cone?
Use πrl only, as there is no base. A party hat with r = 8 cm and l = 20 cm needs 160π ≈ 502.7 cm² of card.
Is the cone formula on the SAT sheet?
Only the volume, V = ⅓πr²h. College Board’s practice test sheet gives no surface area formula.
Keep exploring
Height or slant height: which one does your child use?
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