How to find the radius of a sphere - ACT Math

Card 0 of 54

Question

The surface area of a sphere is $64\pi$ feet. What is the radius?

Answer

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

$SA=4{\pi}r^2\rightarrow r=\sqrt{\frac{SA}{4\pi}}= \sqrt{\frac{64\pi}{4\pi}}=\sqrt16=4$

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Question

What is the radius of a sphere with a volume of $2304\pi$ ? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$2304\pi=\frac{4}{3}\pi r^3$

Solve for . First, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}2304\pi=\pi r^3$

$1728\pi=\pi r^3$

Now, divide out the $\pi$ :

Using your calculator, you can solve for . Remember, if need be, you can raise to the power of $\frac{1}{3}$ 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.

Compare your answer with the correct one above

Question

The volume of a sphere is $972\pi:cm^3$ . What is the diameter of the sphere? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$972\pi = \frac{4}{3}\pi r^3$

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

$972 = \frac{4}{3} r^3$

Next, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} * 972 =r^3$

Using your calculator, solve for . Recall that you can always use the $\frac{1}{3}$ 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 .

Compare your answer with the correct one above

Question

What is the radius of a sphere with a surface area of $169\pi$ ? Round to the nearest hundredth.

Answer

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

$SA = 4\pi r^2$

For our data, this means:

$169\pi = 4\pi r^2$

Solve for . First, divide by $4\pi$ :

Take the square root of both sides:

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Question

What is the radius of a sphere with a volume of $36\prod \textup{ units}^{3}$ ?

Answer

Given the volume of the sphere, $36\prod \textup{units}^{3}$ , you need to use the formula for volume of a sphere $\left ( \left ( \frac{4}{3} \right )\cdot \prod \cdot r^{3}\right )$ and work backwards to find the radius. I would multiply both sides by $\frac{3}{4}$ to get rid of the $\frac{4}{3}$ in the formula. You then have $27\prod=\prod r^{3}$ . Next, divide both sides by $\prod$ so that all vyou have left is $27=r^{3}$ . Finally take the cube root of , to get units for the radius.

Compare your answer with the correct one above

Question

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?

Answer

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 $D = s\sqrt{3}$ , 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:

$D = s\sqrt3 = 25\sqrt3$

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

$r = \frac{D}{2} = \frac{25\sqrt{3}}{2}$

$r = \frac{25\sqrt{3}}{2}$

Compare your answer with the correct one above

Question

The surface area of a sphere is $64\pi$ feet. What is the radius?

Answer

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

$SA=4{\pi}r^2\rightarrow r=\sqrt{\frac{SA}{4\pi}}= \sqrt{\frac{64\pi}{4\pi}}=\sqrt16=4$

Compare your answer with the correct one above

Question

What is the radius of a sphere with a volume of $2304\pi$ ? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$2304\pi=\frac{4}{3}\pi r^3$

Solve for . First, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}2304\pi=\pi r^3$

$1728\pi=\pi r^3$

Now, divide out the $\pi$ :

Using your calculator, you can solve for . Remember, if need be, you can raise to the power of $\frac{1}{3}$ 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.

Compare your answer with the correct one above

Question

The volume of a sphere is $972\pi:cm^3$ . What is the diameter of the sphere? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$972\pi = \frac{4}{3}\pi r^3$

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

$972 = \frac{4}{3} r^3$

Next, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} * 972 =r^3$

Using your calculator, solve for . Recall that you can always use the $\frac{1}{3}$ 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 .

Compare your answer with the correct one above

Question

What is the radius of a sphere with a surface area of $169\pi$ ? Round to the nearest hundredth.

Answer

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

$SA = 4\pi r^2$

For our data, this means:

$169\pi = 4\pi r^2$

Solve for . First, divide by $4\pi$ :

Take the square root of both sides:

Compare your answer with the correct one above

Question

What is the radius of a sphere with a volume of $36\prod \textup{ units}^{3}$ ?

Answer

Given the volume of the sphere, $36\prod \textup{units}^{3}$ , you need to use the formula for volume of a sphere $\left ( \left ( \frac{4}{3} \right )\cdot \prod \cdot r^{3}\right )$ and work backwards to find the radius. I would multiply both sides by $\frac{3}{4}$ to get rid of the $\frac{4}{3}$ in the formula. You then have $27\prod=\prod r^{3}$ . Next, divide both sides by $\prod$ so that all vyou have left is $27=r^{3}$ . Finally take the cube root of , to get units for the radius.

Compare your answer with the correct one above

Question

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?

Answer

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 $D = s\sqrt{3}$ , 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:

$D = s\sqrt3 = 25\sqrt3$

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

$r = \frac{D}{2} = \frac{25\sqrt{3}}{2}$

$r = \frac{25\sqrt{3}}{2}$

Compare your answer with the correct one above

Question

The surface area of a sphere is $64\pi$ feet. What is the radius?

Answer

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

$SA=4{\pi}r^2\rightarrow r=\sqrt{\frac{SA}{4\pi}}= \sqrt{\frac{64\pi}{4\pi}}=\sqrt16=4$

Compare your answer with the correct one above

Question

What is the radius of a sphere with a volume of $2304\pi$ ? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$2304\pi=\frac{4}{3}\pi r^3$

Solve for . First, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}2304\pi=\pi r^3$

$1728\pi=\pi r^3$

Now, divide out the $\pi$ :

Using your calculator, you can solve for . Remember, if need be, you can raise to the power of $\frac{1}{3}$ 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.

Compare your answer with the correct one above

Question

The volume of a sphere is $972\pi:cm^3$ . What is the diameter of the sphere? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$972\pi = \frac{4}{3}\pi r^3$

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

$972 = \frac{4}{3} r^3$

Next, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} * 972 =r^3$

Using your calculator, solve for . Recall that you can always use the $\frac{1}{3}$ 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 .

Compare your answer with the correct one above

Question

What is the radius of a sphere with a surface area of $169\pi$ ? Round to the nearest hundredth.

Answer

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

$SA = 4\pi r^2$

For our data, this means:

$169\pi = 4\pi r^2$

Solve for . First, divide by $4\pi$ :

Take the square root of both sides:

Compare your answer with the correct one above

Question

What is the radius of a sphere with a volume of $36\prod \textup{ units}^{3}$ ?

Answer

Given the volume of the sphere, $36\prod \textup{units}^{3}$ , you need to use the formula for volume of a sphere $\left ( \left ( \frac{4}{3} \right )\cdot \prod \cdot r^{3}\right )$ and work backwards to find the radius. I would multiply both sides by $\frac{3}{4}$ to get rid of the $\frac{4}{3}$ in the formula. You then have $27\prod=\prod r^{3}$ . Next, divide both sides by $\prod$ so that all vyou have left is $27=r^{3}$ . Finally take the cube root of , to get units for the radius.

Compare your answer with the correct one above

Question

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?

Answer

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 $D = s\sqrt{3}$ , 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:

$D = s\sqrt3 = 25\sqrt3$

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

$r = \frac{D}{2} = \frac{25\sqrt{3}}{2}$

$r = \frac{25\sqrt{3}}{2}$

Compare your answer with the correct one above

Question

The surface area of a sphere is $64\pi$ feet. What is the radius?

Answer

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

$SA=4{\pi}r^2\rightarrow r=\sqrt{\frac{SA}{4\pi}}= \sqrt{\frac{64\pi}{4\pi}}=\sqrt16=4$

Compare your answer with the correct one above

Question

What is the radius of a sphere with a volume of $2304\pi$ ? Round to the nearest hundredth.

Answer

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

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$2304\pi=\frac{4}{3}\pi r^3$

Solve for . First, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}2304\pi=\pi r^3$

$1728\pi=\pi r^3$

Now, divide out the $\pi$ :

Using your calculator, you can solve for . Remember, if need be, you can raise to the power of $\frac{1}{3}$ 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.

Compare your answer with the correct one above

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