# High School Math : How to find the surface area of a cone

## Example Questions

### Example Question #11 : Cones

What is the surface area of a cone with a radius of 4 and a height of 3?

Explanation:

Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

### Example Question #1 : How To Find The Surface Area Of A Cone

The lateral area is twice as big as the base area of a cone.  If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

90π

27π

54π

81π

81π

Explanation:

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r2)

rl = 2r2

l = 2r

From the diagram, we can see that r2 + h2 = l2.  Since h = 9 and l = 2r, some substitution yields

r2 + 92 = (2r)2

r2 + 81 = 4r2

81 = 3r2

27 = r2

B = π(r2) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

### Example Question #1 : How To Find The Surface Area Of A Cone

What is the surface area of a cone with a height of 8 and a base with a radius of 5?

Explanation:

To find the surface area of a cone we must plug in the appropriate numbers into the equation

where is the radius of the base, and is the lateral, or slant height of the cone.

First we must find the area of the circle.

To find the area of the circle we plug in our radius into the equation of a circle which is

This yields .

We then need to know the surface area of the cone shape.

To find this we must use our height and our radius to make a right triangle in order to find the lateral height using Pythagorean’s Theorem.

Pythagorean’s Theorem states

Take the radius and height and plug them into the equation as a and b to yield

First square the numbers

After squaring the numbers add them together

Once you have the sum, square root both sides

After calculating we find our length is

Then plug the length into the second portion of our surface area equation above to get

Then add the area of the circle with the conical area to find the surface area of the entire figure

### Example Question #1 : How To Find The Surface Area Of A Cone

What is the surface area of a cone with a radius of 6 in and a height of 8 in?

36π in2

112π in2

96π in2

66π in2

60π in2

96π in2

Explanation:

Find the slant height of the cone using the Pythagorean theorem:  r2 + h2 = s2 resulting in 62 + 82 = s2 leading to s2 = 100 or s = 10 in

SA = πrs + πr2 = π(6)(10) + π(6)2 = 60π + 36π = 96π in2

60π in2 is the area of the cone without the base.

36π in2 is the area of the base only.

### Example Question #2 : How To Find The Surface Area Of A Cone

Find the surface area of a cone that has a radius of 12 and a slant height of 15.

Explanation:

The standard equation to find the surface area of a cone is

where  denotes the slant height of the cone, and  denotes the radius.

Plug in the given values for  and  to find the answer:

### Example Question #3 : How To Find The Surface Area Of A Cone

Find the surface area of the following cone.

Explanation:

The formula for the surface area of a cone is:

where  is the radius of the cone and  is the slant height of the cone.

Plugging in our values, we get:

### Example Question #4 : How To Find The Surface Area Of A Cone

Find the surface area of the following cone.

Explanation:

The formula for the surface area of a cone is:

Use the Pythagorean Theorem to find the length of the radius:

Plugging in our values, we get:

### Example Question #5 : How To Find The Surface Area Of A Cone

Find the surface area of the following half cone.

Explanation:

The formula for the surface area of the half cone is:

Where  is the radius,  is the slant height, and  is the height of the cone.

Use the Pythagorean Theorem to find the height of the cone:

Plugging in our values, we get: