Spheres

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If the diameter of a sphere is , what is the surface area?

$225\pi$

$900\pi$

$42\pi$

$500\pi$

$56.25\pi$

Explanation

The radius is half the length of a diameter.

$r=\frac{d}{2}=\frac{15}{2}=7.5$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi r^2 = 4\pi(7.5)^2= 225\pi$

Find the surface area of a hemisphere if it has a radius of .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (7)^2=461.81$

Make sure to round to places after the decimal.

If the volume of a sphere is $120\ ft^{3}$ , what is the approximate length of its diameter?

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

$6.12\ ft$

$4.48\ ft$

$3.06\ ft$

$8.96\ ft$

$32.25\ ft$

Explanation

The correct answer is 6.12 ft.

Plug the value of into the equation so that

$120=\frac{4}{3}\pi r^{3}$

Multiply both sides by 3 to get

$360=4\pi r^{3}$

Then divide both sides by $4\pi$ to get

$28.65=r^{3}$

Then take the 3rd root of both sides to get 3.06 ft for the radius. Finally, you have to multiply by 2 on both sides to get the diameter. Thus

What is the volume of a sphere with a diameter of 6 in?

$36\pi\ in^{3}$

$216\pi\ in^{3}$

$108\pi\ in^{3}$

$72\pi\ in^{3}$

$288\pi\ in^{3}$

Explanation

The formula for the volume of a sphere is:

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

where = radius. The diameter is 6 in, so the radius will be 3 in.

If the volume of a sphere is $120\ ft^{3}$ , what is the approximate length of its diameter?

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

$6.12\ ft$

$4.48\ ft$

$3.06\ ft$

$8.96\ ft$

$32.25\ ft$

Explanation

The correct answer is 6.12 ft.

Plug the value of into the equation so that

$120=\frac{4}{3}\pi r^{3}$

Multiply both sides by 3 to get

$360=4\pi r^{3}$

Then divide both sides by $4\pi$ to get

$28.65=r^{3}$

Then take the 3rd root of both sides to get 3.06 ft for the radius. Finally, you have to multiply by 2 on both sides to get the diameter. Thus

Find the surface area for a hemisphere that has a radius of $\frac{1}{2}$ .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (\frac{1}{2})^2=2.36$

Make sure to round to places after the decimal.

If the radius of the sphere is one half, what is the surface area?

$\pi$

$\frac{1}{4}\pi$

$\frac{1}{2}\pi$

$\frac{4}{3}\pi$

$2\pi$

Explanation

Write the formula for the surface area of a sphere, plug in the radius, and find the area.

$A=4\pi r^2 = 4\pi \left(\frac{1}{2}\right)^2 = 4\pi\left(\frac{1}{4}\right)= \pi$

Find the surface area of a sphere if the volume is .

$4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

$\sqrt[3]{9}$

$4\pi\left(\frac{4}{3\pi}\right)^\frac{2}{3}$

$\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

$\sqrt[3]{\frac{3\pi}{4}}$

Explanation

Write the formula for the volume of a sphere and substitute volume to find radius.

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

$1=\frac{4}{3}\pi r^3$

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

$r=\sqrt[3]{\frac{3}{4\pi}}= (\frac{3}{4\pi})^\frac{1}{3}$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi r^2 = 4\pi \left[\left(\frac{3}{4\pi}\right)^\frac{1}{3}\right]^2 = 4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

Find the surface area of a hemisphere that has a radius of $\frac{3}{2}$ .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (\frac{3}{2})^2=21.21$

Make sure to round to places after the decimal.

Find the surface area of a hemisphere that has a radius of .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (3)^2=84.82$

Make sure to round to places after the decimal.

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