How to find radius of a sphere

The radius of a sphere hides inside its absolute roundness. A sphere's radius is the length from the sphere's center to any point on its surface. The radius is an identifying trait, and from it other measurements of the sphere can be calculated, including its circumference, surface area and volume. The formula to determine the volume of a sphere is 4/3π multiplied by r, the radius, cubed, where π, or pi, is a nonterminating and nonrepeating mathematical constant commonly rounded off to 3.1416. Since we know the volume, we can plug in the other numbers to solve for the radius, r.

Multiply the volume by 3. For example, suppose the volume of the sphere is 100 cubic units. Multiplying that amount by 3 equals 300.

Divide this figure by 4π. In this example, dividing 300 by 4π gives a quotient of 23.873.

Calculate the cube root of that number. For this example, the cube root of 23.873 equals 2.879. The radius is 2.879 units.

Let’s try an example where we’re given a point on the surface and the center of the sphere.

Example

Find the equation of the sphere with center ???(1,1,2)??? that passes through the point ???(2,4,6)???.

Since we’re given the center of the sphere in the question, we can plug it into the equation of the sphere immediately.

???(x-1)^2+(y-1)^2+(z-2)^2=r^2???

We’ll find the radius of the sphere using the distance formula, plugging the point on the surface of the sphere in for ???(x_1,y_1,z_1)???, and plugging the center of the sphere in for ???(x_2,y_2,z_2)???.

???D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}???

???r=\sqrt{(2-1)^2+(4-1)^2+(6-2)^2}???

???r=\sqrt{1+9+16}???

???r=\sqrt{26}???

Plugging this into our equation, we get

???(x-1)^2+(y-1)^2+(z-2)^2=\left(\sqrt{26}\right)^2???

???(x-1)^2+(y-1)^2+(z-2)^2=26???

This is the equation of the sphere. We can also write it as

???(x-1)^2+(y-1)^2+(z-2)^2=26???

???x^2-2x+1+y^2-2y+1+z^2-4z+4=26???

???x^2-2x+y^2-2y+z^2-4z=20???

Remember, using the distance formula to find the radius, we’ll always get a value for ???r???. But we need ???r^2??? in the equation of the sphere. So we can either

solve for ???r???, square it, and then substitute for ???r^2??? into the equation, or

solve for ???r???, substitute for ???r??? into the equation, then square it to simplify.

Either way will work, so do the steps in whichever order you prefer.

Let’s try another example when we’re given the expanded form of the equation and we need to find the center and radius.

How to find the center and radius from the equation of the sphere

Example

Find the center and radius of the sphere.

???x^2+2x+y^2-2y+z^2-6z=14???

We know we eventually need to change the equation into the standard form of the equation of a sphere,

???(x-h)^2+(y-k)^2+(z-l)^2=r^2???

In order to do so, we’ll need to complete the square with respect to each variable. Remember that the process of completing the square requires us to use the coefficient on the first degree term. For ???x??? that’s ???2x??? so the coefficient is ???2???; for ???y??? that’s ???-2y??? so the coefficient is ???-2???; for ???z??? that’s ???-6z??? so the coefficient is ???-6???. Completing the square tells us that we’ll divide each of those coefficients by ???2???, and then take the result that we get and square it. These final values will be what we add into (and subtract out of) the equation of the sphere.

With respect to ???x???:

???\frac22=1???, and then ???1^2=1???

With respect to ???y???:

???\frac{-2}{2}=-1???, and then ???(-1)^2=1???

With respect to ???z???:

???\frac{-6}{2}=-3???, and then ???(-3)^2=9???

Adding each of these values into our equation, and subtracting them out again, we get

???\left(x^2+2x+1\right)-1+\left(y^2-2y+1\right)-1+\left(z^2-6z+9\right)-9=14???

???\left(x^2+2x+1\right)+\left(y^2-2y+1\right)+\left(z^2-6z+9\right)=25???

???(x+1)^2+(y-1)^2+(z-3)^2=25???

With our equation in standard form, we can pull out the center point. Remember, the standard form of a circle is ???(x-h)^2+(y-k)^2+(z-l)^2=r^2???, which means that we have to include a negative sign if we have ???x+x_1???, ???y+y_1???, or ???z+z_1???. The center is at ???(-1,1,3)???.

To find the radius, it’s important that we take the square root of the right-hand side, and not just the full value from the right, since the standard form of the equation of a sphere has ???r^2??? on the right-hand side.

???r^2=25???

???r=5???

To summarize our findings, we can say that the sphere has center ???(-1,1,3)??? and radius ???r=5???.

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

Correct answer:

Explanation:

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

What is the radius of a sphere with a volume of 

Correct answer:

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.

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

Correct answer:

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 .

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

Correct answer:

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:

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

Correct answer:

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?

Correct answer:

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:

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