Surface Area of a Cone –

The surface area of a cone is a Higher tier topic that combines circle area, arc length ideas, and Pythagoras' theorem. You need to distinguish between the curved surface area and the total surface area, and between slant height and vertical height.

What Is the Surface Area of a Cone?

A cone has two surfaces: the circular base and the curved lateral surface that wraps around from the base to the apex. The total surface area is the sum of these two parts.

The curved surface area formula πrl comes from the fact that if you "unroll" the curved surface of a cone, it forms a sector of a circle with radius equal to the slant height l. The area of that sector works out to πrl.

The slant height l is the distance measured along the surface from the base edge to the apex. It is different from the vertical (perpendicular) height h. These two heights and the radius form a right-angled triangle, so l² = r² + h².

Key Formulas

Step-by-Step Method

1. Identify the radius r, vertical height h, and slant height l. If the slant height is not given, use Pythagoras: l = √(r² + h²).
2. Calculate the curved surface area: πrl.
3. Calculate the base area: πr².
4. Add them together for the total surface area, or give just the curved surface area if the question asks for it.

Worked Example 1 — Foundation Level

Question: A cone has a base radius of 5 cm and a slant height of 13 cm. Find the total surface area to 1 decimal place.

Working: Curved SA = πrl = π × 5 × 13 = 65π Base area = πr² = π × 5² = 25π Total SA = 65π + 25π = 90π = 282.7 (1 d.p.)

Answer: 282.7 cm²

Worked Example 2 — Higher Level

Question: A cone has a base radius of 6 cm and a vertical height of 8 cm. Find the total surface area to 1 decimal place.

Working: First find the slant height: l = √(r² + h²) = √(36 + 64) = √100 = 10 cm Curved SA = π × 6 × 10 = 60π Base area = π × 6² = 36π Total SA = 60π + 36π = 96π = 301.6 (1 d.p.)

Answer: 301.6 cm²

Worked Example 3 — Exam Style

Question: The curved surface area of a cone is 110 cm². The slant height is 7 cm. Find the base radius to 2 decimal places.

Working: Curved SA = πrl 110 = π × r × 7 110 = 7πr r = 110 ÷ (7π) r = 110 ÷ 21.991… r = 5.00 (2 d.p.)

Answer: 5.00 cm

Common Mistakes

• Confusing slant height and vertical height. The slant height l runs along the surface; the vertical height h is inside the cone from base centre to apex. Always check which one the question provides.
• Forgetting the base. "Surface area" usually means total surface area including the base. If only the curved surface is required, the question will specify "curved surface area."
• Using diameter instead of radius. The formulas use the radius. If the question gives the diameter, halve it first.
• Not using Pythagoras when needed. When the slant height is not given, you must calculate it using l = √(r² + h²) before applying the surface area formula.

Exam Tips

• The curved surface area formula πrl is given on the exam formula sheet — but you still need to remember to add πr² for the total.
• Show the Pythagoras step clearly if you need to find the slant height — this earns method marks.
• If the question asks for an exact answer, leave your answer in terms of π.

Practice Questions

Q1 (Foundation): A cone has radius 4 cm and slant height 9 cm. Find the curved surface area in terms of π.

Q2 (Foundation): A cone has radius 3 cm and slant height 10 cm. Find the total surface area to 1 decimal place.

Q3 (Higher): A cone has radius 5 cm and vertical height 12 cm. Find the total surface area to the nearest whole number.

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Related Topics

• Surface Area
• Volume of 3D Shapes
• Pythagoras' Theorem

Summary

• A cone has a curved surface and a circular base.
• Curved surface area = πrl, where l is the slant height.
• Total surface area = πrl + πr² (curved + base).
• If the slant height is not given, use Pythagoras: l = √(r² + h²).
• Always check whether the question asks for the curved or the total surface area.