To find the surface area of a sphere, we will use the following formula:

$\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 20in.  We also know the diameter is two times the radius.  Therefore, the radius is 10in. 

Knowing this, we can substitute into the formula.  We get

$\displaystyle SA = 4 \cdot \pi \cdot (10\text{in})^2$

$\displaystyle SA = 4 \cdot \pi \cdot 100\text{in}^2$

$\displaystyle SA = 400 \pi \text{ in}^2$

To find the surface area of a sphere, we will use the following formula:

$\displaystyle SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in. So, we get

$\displaystyle SA = 4 \cdot \pi \cdot (6\text{in})^2$

$\displaystyle SA = 4 \cdot \pi \cdot 36\text{in}^2$

$\displaystyle SA = 144\pi \text{ in}^2$

A spherical buoy has a radius of 5 meters. What is the surface area of the buoy?

To find the surface area of a sphere, use the following:

$\displaystyle SA_{sphere}=4\pi r^2$

Plug in our radius and solve!

$\displaystyle SA_{sphere}=4\pi (5m)^2=100\pi m^2$

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $\displaystyle 18 \div 12 = \frac{18}{12} = \frac{3}{2}$ feet

Radius: 4 inches = $\displaystyle 4 \div 12 = \frac{4}{12} = \frac{1}{3}$

The volume of a cone is given by the formula

$\displaystyle V = \frac{1}{3} \pi r^{2} h$

Substitute $\displaystyle r = \frac{1}{3}, h = \frac{3}{2}$:

$\displaystyle V = \frac{1}{3}\cdot \left ( \frac{1}{3} \right ) ^{2}\cdot \frac{3}{2} \cdot \pi$

$\displaystyle V = \frac{1}{3}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{3}{2} \cdot \pi$

$\displaystyle V = \frac{1}{1}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{2} \cdot \pi$

$\displaystyle V = \frac{1}{18} \pi$

A circle with circumference $\displaystyle 6 \pi$ inches has as its radius 

$\displaystyle r = \frac{C }{2\pi }=\frac{6\pi }{2\pi } = 3$ inches.

The area of the base is therefore

$\displaystyle B = \pi r^{2} = \pi \cdot 3^{2} = 9 \pi$ square inches.

To find the volume of the cone, substitute $\displaystyle B = 9 \pi , h = 10$ in the formula for the volume of a cone:

$\displaystyle V = \frac{1}{3} Bh = \frac{1}{3} \cdot 9 \pi \cdot 10 = 30 \pi$ cubic inches

A circle with circumference $\displaystyle 10 \pi$ inches has as its radius 

$\displaystyle r = \frac{C }{2\pi }=\frac{10\pi }{2\pi } = 5$ inches.

The height is also $\displaystyle 5$ inches, so substitute $\displaystyle h = r = 5$ in the volume formula for a cone:

$\displaystyle V = \frac{1}{3} \pi r ^{2}h = \frac{1}{3} \pi \cdot 5 ^{2} \cdot 5 = \frac{125}{3} \pi$ cubic inches

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

Begin by using the formula for volume of a cone:

$\displaystyle V=\pi r^2\frac{h}{3}$

Now, we simply need to plug in our knowns.

We know the height is 30 ft.

We know that the diameter is 10ft, however, we need the radius.

Divide 10 by 2 to get a radius of 5 ft.

Now, let's go....

$\displaystyle V=\pi (5ft)^2\frac{30ft}{3}=\pi25ft^2*10ft=250 \pi ft^3$

To find the volume of a cone, we will use the following formula:

$\displaystyle V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 12in.  We also know the diameter is two times the radius.  Therefore, the radius is 6in.

We know the height of the cone is 6in.

Knowing all of this, we can substitute into the formula.  We get

$\displaystyle V = \pi \cdot (6\text{in})^2 \cdot \frac{6\text{in}}{3}$

$\displaystyle V = \pi \cdot 36\text{in}^2 \cdot 2\text{in}$

$\displaystyle V = \pi \cdot 72\text{in}^3$

$\displaystyle V = 72\pi \text{ in}^3$

To find the volume of a cone, we will use the following formula:

$\displaystyle V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 14in.  We also know the diameter is two times the radius.  Therefore, the radius is 7in.

We know the height of the cone is 9in.

Knowing all of this, we can substitute into the formula.  We get

$\displaystyle V = \pi \cdot (7\text{in})^2 \cdot \frac{9\text{in}}{3}$

$\displaystyle V = \pi \cdot 49\text{in}^2 \cdot 3\text{in}$

$\displaystyle V = \pi \cdot 147\text{in}^3$

$\displaystyle V = 147\pi \text{ in}^3$

To find the volume of a cone, we will use the following formula:

$\displaystyle V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know the height of the cone is 12in. We also know the diameter of the cone is 6in. We know the diameter is two times the radius. Therefore, the radius is 3in.  

So, we get

$\displaystyle V = \pi \cdot (3\text{in})^2 \cdot \frac{12\text{in}}{3}$

$\displaystyle V = \pi \cdot 9\text{in}^2 \cdot 4\text{in}$

$\displaystyle V = \pi \cdot 36\text{in}^3$

$\displaystyle V = 36\pi \text{ in}^3$