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# Heights of Cylinders Given Surface Area

## h = SA/(2πr) - r

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Practice Heights of Cylinders Given Surface Area
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Heights of Cylinders Given Surface Area

What if you know the surface area of a cylinder, but don't know the height? Look at this dilemma.

Candice is going to use modge podge to decorate the outside of a cylindrical canister. She wants to make it into a decorative pencil holder as a gift for her Mom. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Candice's canister has a radius of 3 inches and has a surface area of $207.3 \ in^2$ .

What is the height of the canister?

Use this Concept to learn about how to figure out the height of a cylinder given the radius and surface area.

### Guidance

Sometimes we may be given the surface area of a cylinder and need to find its height. We simply fill the information we know into the surface area formula. This time, instead of solving for $SA$ , we solve for $h$ , the height of the cylinder.

Let’s look at how we can apply this information.

The surface area of a cylinder with a radius of 3 inches is $78 \pi$ square inches. What is the height of the cylinder?

We know that the radius of the cylinder is 3 inches. We also know that the surface area of the cylinder is $78 \pi$ square inches. Sometimes a problem will include pi in the amount because it is more precise. We’ll see that this also makes our calculations easier. All we have to do is put $78 \pi$ into the formula in place of $SA$ . Then we can use the formula to solve for the height, $h$ .

$SA & = 2 \pi r^2 + 2 \pi rh\\78 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\78 \pi & = 2 \pi (9) + 6 \pi h\\78 \pi & = 18 \pi + 6 \pi h\\78 \pi - 18 \pi & = 6 \pi h \qquad \qquad \qquad \text{Subtract} \ 18 \pi \ \text{from both sides.}\\60 \pi & = 6 \pi h\\60 \pi \div 6 \pi & = h \qquad \qquad \quad \qquad \text{Divide both sides by} \ 6 \pi.\\10 \ in. & = h$

We used the formula to find that a cylinder with a radius of 3 inches and a surface area of $78 \pi$ has a height of 10 inches.

We can check our work by putting the height into the formula and solving for surface area.

$SA & = 2 \pi (3^2) + 2 \pi (3) (10)\\SA & = 2 \pi (9) + 2 \pi (30)\\SA & = 18 \pi + 60 \pi\\SA & = 78 \pi$

This is the surface area given in the problem, so our answer is correct. Let’s look at another example.

Now try a few of these on your own.

Find the height in each problem.

#### Example A

$SA = 48 \pi, \ r = 3 \ in$

Solution: $2 \ in$

#### Example B

$SA = 60 \pi, \ r = 2 \ in$

Solution: $5 \ in$

#### Example C

$SA = 80 \pi, \ r = 4 \ m$

Solution: $2 \ m$

Here is the original problem once again.

Candice is going to use modge podge to decorate the outside of a cylindrical canister. She wants to make it into a decorative pencil holder as a gift for her Mom. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Candice's canister has a radius of 3 inches and has a surface area of $207.24 \ in^2$ .

What is the height of the canister?

To figure this out, let's use the formula.

$SA & = 2 \pi r^2 + 2 \pi rh\\66 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\66 \pi & = 2 \pi (9) + 6 \pi h\\66 \pi & = 18 \pi + 6 \pi h\\66 \pi - 18 \pi & = 6 \pi h \qquad \qquad \qquad \text{Subtract} \ 18 \pi \ \text{from both sides.}\\66 \pi & = 6 \pi h\\66 \pi \div 6 \pi & = h \qquad \qquad \quad \qquad \text{Divide both sides by} \ 6 \pi.\\8 \ in. & = h$

The height of the cylinder is $8 \ in$ .

### Vocabulary

Cylinder
a three-dimensional figure with two circular bases.
Surface Area
the measurement of the outside of a three-dimensional figure.
Net
a two-dimensional representation of a three-dimensional figure.

### Guided Practice

Here is one for you to try on your own.

Mrs. Javitz bought a container of oatmeal in the shape of the cylinder. If the container has a radius of 5 cm and a surface area of $170 \pi$ , what is its height?

Once again, we have been given the surface area as a function of pi. That’s no problem we simply put the whole number, $170 \pi$ , in for $SA$ in the formula. We also know that the container has a radius of 5 centimeters, so we can use the formula to solve for $h$ , the height of the container.

$SA & = 2 \pi r^2 + 2 \pi rh\\170 \pi & = 2 \pi (5^2) + 2 \pi (5) h\\170 \pi & = 2 \pi (25) + 10 \pi h\\170 \pi & = 50 \pi + 10 \pi h\\170 \pi - 50 \pi & = 10 \pi h \quad \qquad \qquad \text{Subtract} \ 50 \pi \ \text{from both sides.}\\120 \pi & = 10 \pi h\\120 \pi \div 10 \pi & = h \qquad \qquad \qquad \ \ \text{Divide both sides by} \ 10 \pi.\\12 \ cm & = h$

The oatmeal container must have a height of 12 centimeters.

### Practice

Directions : Find the height of each cylinder given the surface area and one other dimension.

1. $r = 5 \ in$

$SA = 376.8 \ in^2$

2. $r = 6 \ in$

$SA = 527.52 \ in^2$

3. $r = 5.5 \ in$

$SA = 336.8255 \ in^2$

4. $r = 4 \ cm$

$SA = 251.2 \ cm^2$

5. $r = 5 \ m$

$SA = 439.6 \ m^2$

6. $r = 2 \ m$

$SA = 87.92 \ cm^2$

7. $r = 2 \ cm$

$SA = 75.35 \ cm^2$

8. $d = 4 \ m$

$SA = 125.6 \ m^2$

9. $d = 8 \ cm$

$SA = 351.68 \ cm^2$

10. $d = 6 \ cm$

$SA = 282.6 \ cm^2$

11. $r = 3 \ cm$

$SA = 226.08 \ cm^2$

12. $r = 14 \ cm$

$SA = 2637.6 \ cm^2$

13. $r = 18 \ cm$

$SA = 4295.53 \ cm^2$

14. $r = 12 \ cm$

$SA = 1959.36 \ cm^2$

15. $r = 13 \ cm$

$SA = 3347.24 \ cm^2$

### Vocabulary Language: English

Cylinder

Cylinder

A cylinder is a solid figure with two parallel congruent circular bases.
Net

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.
Surface Area

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.