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

## Example Questions

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### 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?

66π in2

96π in2

112π in2

36π 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 #11 : How To Find The Surface Area Of A Cone

Use the following formula to answer the question.

The slant height of a right circular cone is . The radius is , and the height is . Determine the surface area of the cone.

Explanation:

Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is , and the slant height is

First plug these numbers into the equation provided.

Then simplify by combining like terms.

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

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

Explanation:

The formula for the surface area of a cone with base of radius  and slant height  is

.

The diameter of the base is ; the radius is half this, so

Substitute in the surface area formula:

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

The radius of the base of a cone is ; its slant height is two-thirds of the diameter of that base. Give its surface area in terms of .

Explanation:

The formula for the surface area of a cone with base of radius  and slant height  is

.

The diameter of the base is twice radius , or , and its slant height is two-thirds of this diameter, which is . Substitute this for  in the formula:

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

The radius of the base of a cone is ; its height is twice of the diameter of that base. Give its surface area in terms of .

Explanation:

The formula for the surface area of a cone with base of radius  and slant height  is

.

The base has radius  and diameter . The height is twice the diamter, which is . Its slant height can be calculated using the Pythagorean Theorem:

Substitute  for  in the surface area formula:

### Example Question #6 : How To Find The Surface Area Of A Cone

The height of a cone is ; the diameter of its base is twice the height. Give its surface area in terms of .

Explanation:

The formula for the surface area of a cone with base of radius  and slant height  is

.

The diameter of the base is twice the height, which is ; the radius is half this, which is .

The slant height can be calculated using the Pythagorean Theorem:

Substitute  for  and  for  in the surface area formula:

### Example Question #7 : How To Find The Surface Area Of A Cone

The circumference of the base of a cone is 80; the slant height of the cone is equal to twice the diameter of the base. Give the surface area of the cone (nearest whole number).

Explanation:

The formula for the surface area of a cone with base of radius  and slant height  is

.

The slant height is twice the diameter, or, equivalently, four times the radius, so

and

The radius of the base is the circumference divided by , which is

Substitute:

### Example Question #8 : How To Find The Surface Area Of A Cone

The circumference of the base of a cone is 100; the height of the cone is equal to the diameter of the base. Give the surface area of the cone (nearest whole number).

Explanation:

The formula for the surface area of a cone with base of radius  and slant height  is

.

The diameter of the base is the circumference divided by , which is

This is also the height .

The radius is half this, or

The slant height can be found by way of the Pythagorean Theorem:

Substitute in the surface area formula:

### Example Question #9 : How To Find The Surface Area Of A Cone

A heat shield on a particular satellite takes the form of a cone. If the surface area of the "face" of the cone (not counting the disk on the bottom) is , and the stripe of reflective paint from the tip of the cone is down to the base is   feet long, what is the diameter of the disk in feet? Round  to 3 significant digits. Round your final answer to the nearest foot.

Explanation:

In this problem, we only need to consider the part of the formula for conic surface area that deals with slant height, since that is all the information we have.

We know the formula for the lateral surface area of a cone is , and we know that  is  feet. Plugging in our other values gives us:

Simplify:

Thus, if our radius is approximately  feet, our diameter is approximately  feet.

### Example Question #10 : How To Find The Surface Area Of A Cone

What is the surface area in square units of a cone with radius  units and height  units?

Explanation:

The formula for surface area of a cone is:

, where  is the slant height. Since we know the radius, we can calculate the first part without issue:

The second part requires us to calculate slant height. Since all cones have a right angle created by the base and height perpendicular to the base, we can use the Pythagorean theorem to calculate :

Now, we can complete our formula. Don't forget to add in the circular base.

Thus, our surface area is  square units.

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