how to find the surface area of a cone

Download Article

The surface area of a cone is the sum of the lateral surface area and the base surface area. If you know the radius of the base and the slant height of the cone, you can easily find the total surface area using a standard formula. Sometimes, however, you might have the radius and some other measurement, such as the height or volume of the cone. In these instances, you can use the Pythagorean Theorem and the volume formula to derive the slant height, and thus the surface area of the cone.

1
2
Plug the value of the radius into the formula. This length should be given, or you should be able to measure it. Make sure you substitute for both $r$ variables in the formula.
- For example, if the radius of the base of a cone is 5 cm, your formula will look like this: ${\text{SA}}=(\pi )(5)(s)+(\pi )(5^{2})$ .
3
Plug the value of the slant height into the formula. This length should be given, or you should be able to measure it.
- For example, if the slant height of a cone is 10 cm, your formula will look like this: ${\text{SA}}=(\pi )(5)(10)+(\pi )(5^{2})$ .
4
5
6
Add the lateral surface area and the base area of the cone. This will give you the total surface area of the cone, in square units.
- For example:
  ${\text{SA}}=157+78.5=235.5$
  So, the surface area of a cone with a radius of 5 cm and a slant height of 10 cm is 235.5 square centimeters.

1
2
3

Square the lengths of the radius and height, then add. Remember that squaring a number means to multiply it by itself.
4

Take the square root of each side of the equation. This will give you the length of the hypotenuse of the right triangle, which is equal to the slant height of the cone.^[7]
5
6
Plug all the known values into the formula. The radius should be given, and you already calculated the slant height. Make sure you use the slant height in the surface area formula, not the (perpendicular) height. If you are not using a calculator, use 3.14 for $\pi$
- For example, for a cone with a radius of 5 cm and a slant height of 13 cm, your formula will look like this: ${\text{SA}}=(3.14)(5)(13)+(3.14)(5^{2})$ .
7

Multiply to find the lateral area and the base area. Then, add these products together. The sum will give you the total surface area of the cone in square units.

1
2
Plug the known values into the formula. You should know the volume and the length of the radius. If not, you cannot use this method. If you are not using a calculator, use 3.14 for $\pi$ .
- For example, if you know a cone has a volume of 950 cubic centimeters and a radius of 6 centimeters, your formula will look like this: $950={\frac {1}{3}}(3.14)(6^{2})(h)$ .
3
4

Divide each side by the $h$ h {\displaystyle h} coefficient. This will give you the value of $h$ , which is the perpendicular height of the cone. You will need this information to find the slant height of the cone, which is necessary to know when solving for the surface area.
5

Set up the formula for the Pythagorean Theorem. The formula is $a^{2}+b^{2}=c^{2}$ , where $a$ and $b$ equal the side lengths of a right triangle, and $c$ equals the length of the hypotenuse (the side opposite the right angle).^[12]
6
7

Solve for $c$ c {\displaystyle c} . This will give you the length of the right triangle's hypotenuse, which is also the slant height of the cone.
8
9
Plug all the known values into the formula. Make sure you use the slant height in the surface area formula, not the (perpendicular) height. If you are not using a calculator, use 3.14 for $\pi$
- For example, for a cone with a radius of 6 cm and a slant height of 25.91 cm, your formula will look like this: ${\text{SA}}=(3.14)(6)(25.91)+(3.14)(6^{2})$ .
10

Multiply to find the lateral area and the base area. Then, add these products together. The sum will give you the total surface area of the cone in square units.

Add New Question

Question

What is the area of the base of a cone with a volume of 36 cubic inches and height of 9 inches?

The formula for volume of a cone is 1/3 x Base Area (i.e. area of a circle) x height. Solution: Volume = 1/3 x BA x H 36 = 1/3 x (BA)* x 9 36 x 3 = BA x 9 (we moved the 1/3 to the other side of the equation, hence it reciprocated) 108/9 = BA Base area of the cone is 12 inches.
Question

How do I show that slant height is 2r?

If you're given the radius of the base and the height of the cone, you can do the Pythagorean theorem.
Question

How do I find the radius of the base of a cone given its surface area of 500 pi and height of 15 cm?

Assuming you are given the lateral surface area and the slant height, divide the lateral surface area by the product of pi and the slant height. If instead of the slant height you are given the perpendicular height, use Method 2 above to find the slant height, then multiply the slant height by pi, and divide that product into the lateral surface area to get the radius of the base.
Question

How do I find the radius of a cone?

The question will either give you the radius or the diameter. If it gives you the diameter, divide the diameter by 2 to get the radius.
Question

How do I find the height of a cone if I am only given the surface area of 96 cm2?

You can't do it. You would also have to know the radius.
Question

How can I find the surface of a cone if I have just the slant height without the radius?

As explained above, if you know the slant height, you also need to know the radius to find the surface area.
Question

How would I find out the surface area given the slant height and the height?

You would do SA=SH. Where SA equals surface area, S equals slant height and H equals height.

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Video

The Pythagorean theorem applies to the radius, perpendicular height, and slant height, with the slant height acting as the hypotenuse: (radius)² + (perpendicular height)² = (slant height)².

Thanks for submitting a tip for review!

About This Article

Article SummaryX

To find the surface area of a cone if you know the length of the slant, use the formula (πrs)+πr^2. Put the value of the radius of the circle at the bottom of the cone into the formula where you see an "r" and be sure to square it where necessary. Then, insert the length of the slant into the formula for "s," and multiply the radius, slant, and pi together. Once you have the first part of the equation, multiply pi by the radius squared. To get the total surface area, add the two values together, and be sure to record your answer in units squared! For help finding the surface area of a cone if you know the radius and the perpendicular height, or the radius and the volume, read on!

Did this summary help you?

Thanks to all authors for creating a page that has been read 211,891 times.