Math - How to find the volume of a cylinder

What is the surface area of a cylinder with a radius of 2 cm and a height of 10 cm?

40π cm2

32π cm2

56π cm2

48π cm2

36π cm2

Explanation

SAcylinder = 2πrh + 2πr2 = 2π(2)(10) + 2π(2)2 = 40π + 8π = 48π cm2

The volume of a cylinder is $81\pi$ . If the radius of the cylinder is , what is the surface area of the cylinder?

$72\pi$

$9\pi$

$27\pi$

$3\pi$

$81\pi$

Explanation

The volume of a cylinder is equal to:

$V=\pi r^2h$

Use this formula and the given radius to solve for the height.

$81\pi=\pi(3)^2h$

$81\pi=9\pi(h)$

Now that we know the height, we can solve for the surface area. The surface area of a cylinder is equal to the area of the two bases plus the area of the outer surface. The outer surface can be "unwrapped" to form a rectangle with a height equal to the cylinder height and a base equal to the circumference of the cylinder base. Add the areas of the two bases and this rectangle to find the total area.

$A=2A_{base}+A_{rec}$

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

Use the radius and height to solve.

$A=2\pi (3)^2+2\pi (3)(9)$

$A=18\pi+54\pi$

$A=72\pi$

What is the surface area of a cylinder with a base diameter of and a height of ?

$36\pi in^{2}$

$14\pi in^{2}$

$66\pi in^{2}$

$48\pi in^{2}$

None of the answers

Explanation

Area of a circle $A=\pi r^{2}=\pi(\frac{d}{2})^{2}=\pi(\frac{6}{2})^{2}=\pi(3)^{2}=9\pi in^{2}$

Circumference of a circle $C=\pi d=6\pi in^{2}$

Surface area of a cylinder $SA=2A+hC=2(9\pi)+8(6\pi)=18\pi+48\pi=66\pi in^{2}$

Calculate the volume of a cylinder with a height of six, and a base with a radius of three.

$54\pi$

$18\pi$

Explanation

The volume of a cylinder is give by the equation $\small V=\pi r^2h$ .

In this example, $\small h=6$ and $\small r=3$ .

$V=\pi(3)^2(6)$

$V=\pi(9)(6)=54\pi$

What is the volume of a cylinder with a radius of and a height of ?

$135\pi$

$9\pi$

$9\pi+15$

$96\pi$

$54\pi$

Explanation

When thinking of a 3D figure, think of it as a stack of something. In this case, a cylinder is a stack of a circles.

The volume will be the area of that base circle times the height of the cylinder. Mathematically that would be $V=(\pi r^2)*h$ .

Plug in our given values and solve.

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

$V=(\pi (3)^2)*15$

$V=9\pi*15$

$V=135\pi$

Find the volume of the following cylinder.

Cylinder

$144 \pi m^3$

$134 \pi m^3$

$124 \pi m^3$

$154 \pi m^3$

$164 \pi m^3$

Explanation

The formula for the volume of a cylinder is:

$V=(\pi r^2)(h)$

Where is the radius of the base and is the height of the cylinder

Plugging in our values, we get:

$V=\pi (4m)^2(9m)$

$V = 144 \pi m^3$

Find the volume of the following cylinder.

Cylinder

$539 \pi m^3$

$545 \pi m^3$

$530 \pi m^3$

$521 \pi m^3$

$556 \pi m^3$

Explanation

The formula for the volume of a cylinder is:

$V = \pi r^2 h$

where is the radius of the base and is the length of the height.

Plugging in our values, we get:

$V = \pi (7m)^2 (11m)$

$V = 539 \pi m^3$

A balloon, in the shape of a sphere, is filled completely with water. The surface area of the filled balloon is $144\pi$ . If all of the balloon's water was emptied into a cylindrical cup of base radius 4, how high would the water level be?

$4\pi$

Explanation

The surface area of a sphere is given by the formula

$S (sphere) = 4\pi*r^2$ , which here equals $144\pi$ . So

$4\pi*r^2 = 144\pi$

To find out what the water level would be, we need to know how much water is in the balloon. So we need to find the volume of the balloon.

The amount of water that is in the balloon is

$V(sphere) = \frac{4\pi*r^3}{3}$

= $\frac{4\pi*6^3}{3}$

$288\pi$

The cylindrical column of water will have this volume, and will have base radius 4. Using the formula for volume of a cylinder, we can solve for the height of the column of water.

$V(cylinder) = \pi*r^2*h$ , where is the base radius 4 in this case.

$288\pi = \pi*(4^2)*h$

Find the volume of a cylinder given that its height and radius are 4 and 11, respectively.

$484\pi$

$121\pi$

$144\pi$

$225\pi$

$312\pi$

Explanation

The standard equation to find the volume of a cylinder is

$V=\pi hr^2$

where denotes height and denotes radius.

Plug in the given values for height and radius to find the volume of the cylinder:

$V=\pi \cdot 4(11)^2=4\cdot 121\cdot\pi =484\pi$

A circle has a circumference of $4\pi$ and it is used as the base of a cylinder. The cylinder has a surface area of $16\pi$ . Find the volume of the cylinder.

$8\pi$

$6\pi$

$4\pi$

$10\pi$

$2\pi$

Explanation

Using the circumference, we can find the radius of the circle. The equation for the circumference is $2\pi r$ ; therefore, the radius is 2.

Now we can find the area of the circle using $\pi r^{2}$ . The area is $4\pi$ .

Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder: $2\cdot 4\pi +4\pi h=16\pi$ , where $h$ is the height of the cylinder.