ISEE Upper Level Quantitative Reasoning - How to find the surface area of a cylinder

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:

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

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

$A = \pi * r^2$

For our problem, this is:

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

You need to double this for the two bases:

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

$2 * \pi * r * h$

For our problem, this is:

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

Therefore, the total surface area is:

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

If a cylinder has a height of 6 mm and a radius of 12 mm, what is its surface area?

$432 \pi mm^2$

$288 \pi mm^2$

$144 \pi mm^2$

Explanation

If a cylinder has a height of 6 mm and a radius of 12 mm, what is its surface area?

Find surface area with the following formula:

$A=2\pi r h +2\pi r^2$

This works because we are adding the area of the two bases to the area of the side.

Plug in and simplify:

$A=2\pi (12mm)(6mm) +2\pi (12mm)^2=144\pi mm^2+288\pi mm^2=432 \pi mm^2$

Find the volume of a cylinder with a diameter of 14cm and a height of 8cm.

$392\pi \text{ cm}^3$

$112\pi \text{ cm}^3$

$896\pi \text{ cm}^3$

$976\pi \text{ cm}^3$

$56\pi \text{ cm}^3$

Explanation

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

$V = \pi r^2 h$

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

Now, we know the diameter of the cylinder is 14cm. We also know that the diameter is two times the radius. Therefore, the radius is 7cm.

We know the height of the cylinder is 8cm.

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

$V = \pi \cdot (7\text{cm})^2 \cdot 8\text{cm}$

$V = \pi \cdot 49\text{cm}^2 \cdot 8\text{cm}$

$V = \pi \cdot 392\text{cm}^3$

$V = 392\pi \text{ cm}^3$

The volume of a cylinder with height of is $275\pi$ . What is its surface area?

$160\pi$

$55\pi$

$135\pi$

$66\pi$

$82\pi$

Explanation

To begin, we must solve for the radius of this cylinder. Recall that the equation of for the volume of a cylinder is:

$V = \pi * r^2 * h$

For our values this is:

$275\pi = \pi * r^2 * 11$

Solving for , we get:

Hence,

Now, 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:

$\pi * 5^2 = \pi * 25$