Surface Area of Cones
A cone has a circular base and a curved surface. The curved surface area is \(\pi rl\), where \(l\) is the slant height, so a closed cone has surface area \(\pi rl+\pi r^2\). In NSW Year 10 Advanced maths Pythagoras' theorem links the radius, height and slant height, and you also make a cone by rolling up a sector of a circle (NESA outcome MA5‑ARE‑P‑01).
You will find total and curved surface areas, work backwards from an area to the radius or slant height, and find the perpendicular height. You will find the base radius of a cone made from a sector, and the sector angle for a paper cup. Practical questions cost the canvas for a teepee and find the area of a lampshade. You will also test the claim that doubling the dimensions doubles the surface area. It is part of Year 10 Advanced Measurement.
Every question with a fully worked solution.
- Surface Area of Cones - Video - Surface Area of a Cone Watch
Theory
The surface area of a cone is the curved surface plus the circular base: \(\text{SA} = \pi rl + \pi r^2\), where \(l\) is the slant height. This Year 10 Advanced maths page covers finding \(l\) with Pythagoras' theorem, rolling a sector into a cone, working backwards to the radius, and cones with the top cut off.
A right cone has a circular base and a curved surface that narrows to the apex, directly above the centre of the base.
Three lengths describe it: the radius \(r\) of the base, the perpendicular height \(h\) from the apex to the centre, and the slant height \(l\) from the apex down the side to the edge of the base. They form a right-angled triangle with \(l\) as the hypotenuse, so \(l\) is always longer than \(h\) and than \(r\).
Cut the curved surface along a slant line and flatten it: you get a sector of a circle. Its radius is \(l\) and its arc is the circumference of the base, \(2\pi r\). Its area is \(\pi rl\).
The total surface area of a solid cone adds the base, \(\pi r^2\). An open cone (party hat, cup, funnel, teepee with no floor) has the curved surface only.
For a cone with radius \(r\) and slant height \(l\):
What to find first:
| Question gives | First find | Using |
|---|---|---|
| \(r\) and \(h\) | \(l\) | \(l^2 = r^2 + h^2\) |
| \(r\) and \(l\) | \(h\) | \(h^2 = l^2 - r^2\) |
| base circumference \(C\) | \(r\), or use \(\pi r = \dfrac{C}{2}\) directly | \(2\pi r = C\) |
| area of the base | \(r\) | \(\pi r^2 = \text{area}\) |
| curved area and \(r\) | \(l\) | \(\pi rl = \text{curved area}\) |
| a sector, radius \(R\), angle \(\theta\) | \(r\) and \(l\) | \(l = R\), \(\dfrac{\theta}{360^\circ} \times 2\pi R = 2\pi r\) |
How to find the surface area of a cone
- Decide which surfaces are there: curved only (open), or curved plus base (solid).
- Find \(r\) and \(l\). Halve a diameter; use Pythagoras' theorem if you have \(h\) instead of \(l\).
- Substitute into \(\pi rl\) and \(\pi r^2\), and add the \(\pi\) terms.
- Round once at the end, or leave the answer in terms of \(\pi\).
Working backwards and harder questions
- Unknown slant height: divide by \(\pi\), subtract the base if you were given the total, and divide by \(r\). This is linear.
- Unknown radius: \(\pi rl + \pi r^2 = \text{SA}\) is a quadratic in \(r\). Factorise (or use the quadratic formula) and keep the positive root.
- One length in terms of another (such as \(l = 3r\)): substitute first, simplify to one term, then solve.
- Lengths given as expressions: \(r^2 + h^2 = l^2\) still holds; expand and solve the equation.
- Scaling: if every length is multiplied by \(k\), the surface area is multiplied by \(k^2\) (doubling gives \(4\) times the area).
- Materials and cost: add any extra percentage for seams before multiplying by the price.
The radius is \(12\) cm. The formula needs the slant height, so use Pythagoras' theorem first.
| \(l^2\) | \(=\) | \(12^2 + 5^2 = 169\) |
| \(l\) | \(=\) | \(13\text{ cm}\) |
| \(\text{SA}\) | \(=\) | \(\pi rl + \pi r^2\) |
| \(\) | \(=\) | \(\pi \times 12 \times 13 + \pi \times 12^2\) |
| \(\) | \(=\) | \(156\pi + 144\pi\) |
| \(\) | \(=\) | \(\mathbf{300\pi}\text{ cm}^2\) |
| \(\) | \(=\) | \(942.4777\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{942.48}\text{ cm}^2\) |
(i) the radius of the cone
(ii) its exact perpendicular height
(iii) the exact curved surface area.
(i) The arc of the sector becomes the circumference of the base:
| \(\text{arc}\) | \(=\) | \(\dfrac{240}{360} \times 2\pi \times 15 = 20\pi\) |
| \(2\pi r\) | \(=\) | \(20\pi\) |
| \(r\) | \(=\) | \(\mathbf{10}\text{ cm}\) |
(ii) The sector's radius is the slant height, \(l = 15\):
| \(h^2\) | \(=\) | \(15^2 - 10^2 = 125\) |
| \(h\) | \(=\) | \(\sqrt{25 \times 5} = \mathbf{5\sqrt{5}}\text{ cm}\) |
(iii) The curved surface is the sector itself:
| \(A\) | \(=\) | \(\pi rl = \pi \times 10 \times 15\) |
| \(\) | \(=\) | \(\mathbf{150\pi}\text{ cm}^2\) |
Check: the sector's area is \(\dfrac{240}{360} \times \pi \times 15^2 = 150\pi\) ✓
Substitute \(l = 15\) and divide every term by \(\pi\). The radius appears squared, so this is a quadratic.
| \(\pi r(15) + \pi r^2\) | \(=\) | \(154\pi\) |
| \(r^2 + 15r - 154\) | \(=\) | \(0\) |
| \((r + 22)(r - 7)\) | \(=\) | \(0\) |
| \(r\) | \(=\) | \(-22 \text{ or } r = 7\) |
A radius is positive, so \(r = \mathbf{7}\) cm. Now the height:
| \(h^2\) | \(=\) | \(15^2 - 7^2 = 176\) |
| \(h\) | \(=\) | \(\sqrt{16 \times 11} = \mathbf{4\sqrt{11}}\text{ cm}\) |
Check: \(\pi(7)(15) + \pi(7)^2 = 105\pi + 49\pi = 154\pi\) ✓. If a quadratic does not factorise, use the quadratic formula.
Complete the cone. The small cut-off cone (slant height \(s\)) and the full cone (slant height \(s + 16\)) are similar, so slant height \(\div\) radius is the same:
| \(\dfrac{s}{4}\) | \(=\) | \(\dfrac{s + 16}{12}\) |
| \(12s\) | \(=\) | \(4s + 64\) |
| \(s\) | \(=\) | \(8\) |
Fabric = full curved surface \(-\) small curved surface:
| \(A\) | \(=\) | \(\pi \times 12 \times 24 - \pi \times 4 \times 8\) |
| \(\) | \(=\) | \(288\pi - 32\pi = 256\pi\) |
| \(\) | \(=\) | \(804.2477\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{804}\text{ cm}^2\) |
Common pitfalls
Frequently asked questions
What is the formula for the surface area of a cone?
The total surface area of a solid cone is \(\text{SA} = \pi rl + \pi r^2\), where \(r\) is the radius and \(l\) is the slant height. The curved surface alone is \(\pi rl\).
How do you find the slant height of a cone?
Use Pythagoras' theorem. The radius and the perpendicular height are the two shorter sides and the slant height is the hypotenuse: \(l^2 = r^2 + h^2\).
Why is the curved surface area of a cone pi r l?
Flattened out, the curved surface is a sector of radius \(l\). Its arc is \(2\pi r\), the fraction \(\dfrac{r}{l}\) of the full circle, so its area is \(\dfrac{r}{l} \times \pi l^2 = \pi rl\).
How do you make a cone from a sector?
Join the two straight edges of the sector. The radius of the sector becomes the slant height of the cone, and the arc length becomes the circumference of the base. Set the arc length equal to 2 pi r to find the cone's radius.
How do you find the surface area of a cone with the top cut off?
Complete the full cone. Use similar triangles to find the slant height of the small cone that was removed, then subtract its curved area from the curved area of the full cone. Add the circles at each end only if they are part of the object.