# ACT Math : How to find the radius of a sphere

## Example Questions

### Example Question #1 : How To Find The Radius Of A Sphere

The surface area of a sphere is  feet. What is the radius?

Explanation:

Solve the equaiton for the surface area of a sphere for the radius and plug in the values:

### Example Question #2 : How To Find The Radius Of A Sphere

What is the radius of a sphere with a volume of  ?  Round to the nearest hundredth.

Explanation:

Recall that the equation for the volume of a sphere is:

For our data, we know:

Solve for . First, multiply both sides by :

Now, divide out the :

Using your calculator, you can solve for . Remember, if need be, you can raise  to the power of  if your calculator does not have a variable-root button.

This gives you:

If you get something like , just round up. This is a rounding issue with some calculators.

### Example Question #3 : How To Find The Radius Of A Sphere

The volume of a sphere is . What is the diameter of the sphere? Round to the nearest hundredth.

Explanation:

Recall that the equation for the volume of a sphere is:

For our data, we know:

Solve for . Begin by dividing out the  from both sides:

Next, multiply both sides by :

Using your calculator, solve for . Recall that you can always use the  power if you don't have a variable-root button.

You should get:

If you get , just round up to . This is a general rounding problem with calculators. Since you are looking for the diameter, you must double this to .

### Example Question #1 : Spheres

What is the radius of a sphere with a surface area of  ?  Round to the nearest hundredth.

Explanation:

Recall that the surface area of a sphere is found by the equation:

For our data, this means:

Solve for . First, divide by :

Take the square root of both sides:

### Example Question #5 : How To Find The Radius Of A Sphere

What is the radius of a sphere with a volume of ?

Explanation:

Given the volume of the sphere, , you need to use the formula for volume of a sphere  and work backwards to find the radius. I would multiply both sides by  to get rid of the  in the formula. You then have . Next, divide both sides by  so that all vyou have left is . Finally take the cube root of , to get  units for the radius.

### Example Question #6 : How To Find The Radius Of A Sphere

A cube with sides of  is circumscribed by a sphere, such that all eight vertices of the cube are tangent to the sphere. What is the sphere's radius?

Explanation:

Solving this problem requires recognizing that since the cube is circumscribed by the sphere, both solids share the same center. Now it is just a matter of finding the diagonal of the cube, which will double as the diameter of the sphere (by definition, any straight line which passes through the center of the sphere). The formula for the diagonal of a cube is , where  is the length of the side of a cube. (This occurs because you must use the Pythagorean theorem once for each 2-dimensional "corner" you travel to find the diagonal for a 3-dimensional shape, but for the ACT it's much faster to memorize the formula.)

In this case:

Since the radius is half the diameter, divide the result in half: