Solid Geometry

Help Questions

Math › Solid Geometry

Questions 1 - 10

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 surface area of a cylinder of height in, with a radius of in?

$152\pi$

$122\pi$

$120\pi$

$136\pi$

$240\pi$

Explanation

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * r^2$

For our problem, this is:

$\pi * 4^2 = \pi * 16$

You need to double this for the two bases:

$2 * 16 * \pi = 32 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 4 * 15 = 120 * \pi$

Therefore, the total surface area is:

$32\pi + 120\pi = 152\pi$

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 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.

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

If the radius of a sphere is , what is the sphere's surface area?

$4\pi:units$

$8\pi:units$

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

$\frac{16}{9}\pi:units$

$2\pi:units$

Explanation

Write the formula for surface area of a sphere:

$A=4\pi r^2$

Plug in the value of radius and solve for the surface area:

$A=4\pi (1)^2 = 4\pi$

Find the surface area of a hemisphere with 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.

$\text{Surface Area of Hemisphere}=3\pi (11)^2=1140.40$

Make sure to round to places after the decimal.

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 sphere with a circumference of $\small \small 10\pi,cm$ .

$\small \small 100\pi,cm^2$

$\small \small 20\pi,cm^2$

$\small \small 400\pi,cm^2$

$\small \small 25\pi,cm^2$

$\small 200\pi,cm^2$

Explanation

The formula for the surface area of a sphere is simply

$\small A=4\pi r^2$

However, the problem is that we are given the circumference rather than the radius. Nonetheless, this isn't too big of an obstacle, as the formula for circumference in terms of radius is simply.

$\small C=2\pi r$