Geometry - How to find the surface area of a cylinder

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(7)^2+2\pi(7)(14)=294\pi=923.63$

Make sure to round to places after the decimal point.

Given a cylinder with radius of 5cm and a height of 10cm, what is the surface area of the entire cylinder?

$150\pi ; cm^2$

$100\pi ; cm^2$

$175\pi ; cm^2$

$200\pi; cm^2$

$50\pi; cm^2$

Explanation

The surface area of the whole cylinder = (2 * area of circle) + lateral area

Think of the lateral area as the paper label on a can; It wraps around the outside of the can while leaving the top and bottom untouched. The area of the circle, times 2, is to account for the top and the bottom of the cylinder.

Area of a circle = $\pi r^2$

So the area of the circle = $25\pi$ , and since there are two circles we have

Now for the lateral area. Notice how if we have a can with a paper label, we can take the label, cut it, and unroll it from the can. In this way, our label now looks like a rectangle with a

height = height and the

width = circumference of the circle.

Circumference = $2\pi r$

So our rectangle is going to have a height of 10 and a width of 10 $\pi$ . So the lateral area =

$10\cdot 10\pi =100\pi$

So the total surface area =

$50\pi +100\pi =150\pi \ cm^{2}$

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(2)^2+2\pi(2)(15)=68\pi=213.63$

Make sure to round to places after the decimal point.

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(20)=682\pi=2142.57$

Make sure to round to places after the decimal point.

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(25)=792\pi=2488.14$

Make sure to round to places after the decimal point.

How many gallon cans of paint must be purchased in order to put a single coat of paint over the surface of a cylindrical water tank if the tank is 75 feet high and 25 feet in radius, and each gallon can of paint covers 350 square feet?

Assume that there is a side, a top, and a bottom to be painted.

Explanation

First, use the formula $A = 2\pi r (r+h)$ to find the surface area of the tank in feet.

$A = 2\pi r (r+h) =A = 2\pi \cdot 25 \cdot (25+75) \approx 15,708$

Now divide by 350, remembering to round up.

$15,708 \div 350 \approx 44.9$

45 cans of paint need to be purchased.

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(23)=748\pi=2349.91$

Make sure to round to places after the decimal point.

Inscribed

The above diagram shows a sphere inscribed inside a cylinder.

The sphere has a surface area of 100. Give the surface area of the cylinder.

$133\frac{1}{3}$

$66\frac{2}{3}$

Explanation

Let be the radius of the sphere. Then the radius of the base of the cylinder is also , and the height of the cylinder is .

The surface area of the cylinder is

$A_{1} = 2\pi r (r+h)$

which, after substituting, is

$A_{1} = 2\pi r (r+2r)$

$A_{1} =2 \pi r (3r)$

$A_{1} =2\cdot 3 \cdot \pi r \cdot r$

$A_{1} = 6 \pi r^{2}$

The surface area of the sphere is

$A_{2} = 4 \pi r^{2}$

Therefore, the ratio of the former to the latter is

$\frac{A_{1} }{A_{2}}= \frac{6 \pi r^{2}}{4\pi r^{2}} = \frac{6}{4} = \frac{3}{2}$

and

$A_{1} = \frac{3}{2} A_{2}$

That is, the cylinder has surface area $\frac{3}{2}$ that of the sphere. Therefore, this area is

$A_{1} = \frac{3}{2} \cdot 100 = 150$ .

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(2)^2+2\pi(2)(5)=28\pi=87.96$

Make sure to round to places after the decimal point.

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(3)^2+2\pi(3)(16)=114\pi=358.14$

Make sure to round to places after the decimal point.

How to find the surface area of a cylinder

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation