Surface area of a cone
A closed cone has two parts: the flat circular base and the curved sloping surface. The curved part, called the curved surface area, uses the slant height \(l\), the distance from the rim up the slope to the apex:
Add the base circle \(\pi r^2\) to get the total surface area of a solid cone:
- Slant height
- The distance measured along the sloping surface of a cone, from a point on the base circle to the apex. It is longer than the perpendicular height.
Finding the slant height
Often you are given the radius \(r\) and the perpendicular height \(h\), not the slant \(l\). These three form a right-angled triangle inside the cone, so Pythagoras links them: \(l^2 = r^2 + h^2\), which gives \(l = \sqrt{r^2 + h^2}\).
Slant for surface, perpendicular for volume
A cone question hands you two heights that are easy to swap by mistake. Surface area uses the slant height \(l\); volume uses the perpendicular height \(h\). If you only have one, use \(l^2 = r^2 + h^2\) to find the other before you start.
Surface area of a sphere
A sphere has a single smooth surface with no edges or corners. Its area depends only on the radius:
Here the radius is squared, so surface area grows in square units, in step with an area rather than a volume. A neat fact worth remembering: a sphere's surface area is exactly four times the area of the flat circle through its middle.
Pyramids from the net
For a pyramid there is no single tidy formula, so unfold it into its net and add the pieces. A square-based pyramid opens out into one square base plus four identical triangles, one for each edge of the base. Each triangle uses the pyramid's slant height as its own height. So the total is the base area plus the four triangle areas.
A solid cone has radius 6 cm and slant height 10 cm. Find its total surface area, to 1 decimal place.
Where this is assessed
Knowing which formula fits each solid is Criterion A (Knowing and understanding). Laying out the net or the separate faces with correct square units is Criterion C (Communicating). Working out the material to make a tent or an ice-cream cone is Criterion D (Applying maths in context). Decide first whether the question wants the curved surface only or the total including the base.
Check yourself
1. Find the surface area of a sphere with radius 7 cm (to 1 d.p.). +
\(\text{Surface area} = 4\pi r^2 = 4\pi \times 7^2 = 4\pi \times 49 = 196\pi = 615.75\ldots \approx \mathbf{615.8}\) cm\(^2\).
2. Find the curved surface area of a cone with radius 4 cm and slant height 9 cm (to 1 d.p.). +
Curved surface area \(= \pi r l = \pi \times 4 \times 9 = 36\pi = 113.09\ldots \approx \mathbf{113.1}\) cm\(^2\).
3. A cone has radius 5 cm and perpendicular height 12 cm. Find its slant height, then its curved surface area (to 1 d.p.). +
Slant height: \(l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\) cm. Curved surface area \(= \pi r l = \pi \times 5 \times 13 = 65\pi = 204.20\ldots \approx \mathbf{204.2}\) cm\(^2\).
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