How to Find the Surface Area of a Cone

Two numbers, added. The awkward part is that the second one wants a length most cones are not labelled with. Drawings give you the height, the formula wants the slant, and the gap between those two is where nearly every wrong answer comes from.

The slant is not the height

Stand a cone up and there is a right triangle hiding inside it. One leg is the radius, lying flat. The other is the height, standing straight up the middle. The hypotenuse runs up the outside from the rim to the tip, and that is the slant height, written l.

So the slant comes from Pythagoras: l equals the square root of r squared plus h squared. For a radius of 6 and a height of 8 that is √(36 + 64), which is 10. Being a hypotenuse, it is always longer than the height, and a squat wide cone exaggerates the difference far more than a tall narrow one.

Where π r l actually comes from

The base is obvious. The sloping surface is the part that looks like it was invented, and it is worth five minutes because the explanation is genuinely satisfying. Cut the cone from rim to tip and roll it flat. What you get is not a triangle and not a circle, it is a sector: a slice of a circle whose radius is the slant height.

The curved edge of that sector used to be the rim of the base, so its length is the base circumference, 2πr, which here is 37.70. A sector’s area is half its arc times its radius, so the sloping surface is half of 37.70 times 10, which is 188.50. Write that in symbols and you get half of 2πr times l, and the halves and twos cancel to leave πrl. That is the whole derivation, and it needs no calculus.

A worked check you can repeat on paper

That last row is a coincidence and a good one to notice. The volume, π r² h over 3, is 96π. The surface area, π r (r + l), is also 96π. Identical digits, and they mean completely different things: 301.59 square units of material against 301.59 cubic units of space. It only happens for this particular cone, and it is a useful reminder that matching numbers prove nothing on their own.

The cost of using the height by mistake

Twenty percent is not a rounding error, it is a missing tin of paint. And because the total still looks like a sensible number, nothing about the answer announces that it is wrong. The only defence is the habit of computing the slant first, as a separate step, before touching the surface area formula at all.

Mistakes that catch people out

• Using the height in place of the slant. The formula wants l, the distance up the outside. Using h costs 20 percent of the sloping surface in the worked example.
• Including a base that is not there. A party hat, a funnel and a roof cone have no bottom. For those, the answer is the lateral surface πrl on its own, and adding πr² overstates it by 113.10 here.
• Doubling something. There is no 2 anywhere in this formula. Nothing on a cone comes in matching pairs, unlike a prism where everything does.
• Squaring the slant. The base term squares the radius, the slope term does not square anything. It is πrl, one radius and one slant, both to the first power.
• Confusing it with volume. Volume divides by three, surface area never does. If a 3 has appeared in your surface area working, something has gone wrong.

Try these

Common questions

What is the formula for the surface area of a cone?

π r² for the base plus π r l for the sloping surface, which factors to π r (r + l). A cone of radius 6 and slant 10 gives π times 6 times 16, which is 301.59 square units.

What is the slant height, and how do you find it?

The distance from the rim up the outside to the tip. It is the hypotenuse of the radius and the vertical height, so l equals the square root of r squared plus h squared. A radius of 6 with a height of 8 gives a slant of 10.

What is the lateral surface area of a cone?

The sloping part on its own, π r l, without the base. Use it for anything open at the bottom, such as a funnel or a conical roof. For radius 6 and slant 10 it is 188.50 square units.

Why is the lateral surface π r l?

Unroll it and it becomes a sector of radius l whose arc is the base circumference, 2πr. A sector’s area is half its arc times its radius, so that is half of 2πr times l, and the 2 cancels the half to leave πrl.

What happens if I use the height instead of the slant?

The answer comes out too small, because the slant is always the longer of the two. In the worked example the sloping surface drops from 188.50 to 150.80, which is 20 percent low, and the total falls 12.5 percent short.

What angle is the sector when a cone is flattened?

360 times r over l. The worked cone has 6 over 10, so 216 degrees. A tall narrow cone gives a thin sector and a wide flat one gives a sector close to a full circle.

Related lessons and calculators

• Circumference of a circle, the 2πr that becomes the curved edge of the sector when the cone is unrolled.
• Surface area of a triangular prism, the same ends and a wrap thinking applied to a shape with flat sides.
• Surface area of a rectangular prism, where nets are introduced from the simplest case.
• Area formulas for every common shape, including the sector formula this derivation leans on.
• Volume of 3D shapes calculator, which also handles a cut off cone, the frustum.
• the National Council of Teachers of Mathematics, for how solids and nets are sequenced in the curriculum.
• Radius of a cylinder, the same reverse thinking applied to the solid with straight sides.
• Volume of a pyramid, which shares its one third rule with the cone and wants the vertical height instead.
• Area of a sector, the formula this derivation leans on, worked through on its own terms.
• Volume of a cone, the same cone measured for what it holds, where the vertical height takes over from the slant.
• Surface area of a sphere, the other curved surface that yields to an unrolling argument.
• Height of a cone, turning the slant into the vertical height that the volume formula actually wants.