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A cylinder is a three-dimensional shape that has two flat ends that are connected by a curved surface. We can find the area of a cylinder by adding the area of the flat surfaces and the area of the curved surface. Learn how to calculate the surface area of a cylinder using a simple formula and check out the solved examples to get a better understanding of the steps involved in the calculation....Read MoreRead Less

A cylinder is a three-dimensional solid. It consists of two congruent plane circular surfaces called **bases** of cylinder and one curved **lateral surface**. There may be different types of cylinders like oblique, elliptical, rectangular, right circular, etc. However, generally we refer to the right circular cylinder when we use the term cylinder. The cylinder is called **right** circular cylinder when one of its bases lies exactly above the other base. Otherwise, if the bases are not exactly above each other, then the cylinder is called an oblique cylinder.

The surface area of a cylinder is the amount of space covered by the plane surfaces of the cylinder. There are two circular planes as the base and a curved lateral surface of a cylinder. The total surface area of the cylinder is the summation of the area of the two circular bases and the area of the curved surface.

There are three types of surface area for a right circular cylinder:

1. The base surface area of cylinder

2. The lateral surface area of cylinder

3. The total surface area of cylinder

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**Formula for the base surface area of a cylinder:**

The base of the cylinder is circular. So, the area of circular base of the cylinder = \(\pi r^2\)

Where,

\(\pi \) is \(\frac{22}{7}\) or 3.14 and “r” is the radius of the cylinder.

**Formula for the lateral surface area of a cylinder:**

If we unfold the lateral surface of the cylinder cutting the surface perpendicular to its base then we can get a rectangle as shown in the figure. The height of the cylinder will be the one side of the rectangle and the circumference of the circular base represents the other side of the rectangle. So, we can find the area of the lateral surface using the formula applying which we can calculate the area of a rectangle.

Lateral surface area of cylinder = 2 \(\pi \) r \(~\times~\)h

Hence the lateral surface area of a cylinder = 2\(\pi \)rh sq. unit

Where,

\(\pi \) is \(\frac{22}{7}\) or 3.14, “r” is the radius of the cylinder and “h” is the height of the cylinder.

**Formula for the total surface area of a cylinder: **

The total surface area of a cylinder is obtained by adding the area of the two bases and the area of the lateral surface of the cylinder. So, the formula for the total surface area of the cylinder is expressed as,

Total surface area of cylinder = area of bases + area of lateral surface area

Total surface area of cylinder = 2\(\pi r^2\) + 2 \(\pi\) rh

T.S.A. of cylinder = 2 \(\pi \)r(r + h) + h Sq. unit

Where,

\(\pi \) is \(\frac{22}{7}\) or 3.14, “r” is the radius and “h” is the height of the cylinder.

Read more on Surface and lateral area 3d figure cylinder formula

**Example 1: **The radius of a cylinder is 8 inches and the height of the same cylinder is 12 inches. Find the total surface area of this cylinder.

(Take the value of \(\pi \) as 3.14)

**Solution:**

Given,

Radius of the cylinder “r” = 8 inches

Height of the cylinder “h” = 12 inches

Surface area = 2\(\pi \)r(r + h)

= 2 × 3.14 \(\times \) 8(8 + 12) [Replace r with 8 and h with 12]

= 2 × 3.14 × 8 × 20 [Simplify]

\(\approx \) 1004.8 [Simplify]

The surface area of the cylinder is 1004.8 square inches.

**Question 2: **Tyler has a water tank. He wants to paint its outer wall. Find the cost of painting the tank if the cost per square inch is $0.40. The radius of the tank is 50 inches and the height of the tank is 100 inches.

**Solution: **

The tank is to be painted outside means the lateral surface should be painted. So, to find the total cost of painting we can find the lateral surface area of the cylinder and multiply that with the unit cost.

2\(\pi\)rh [Lateral surface area of a cylinder formula]

= 2 × 3.14 × 50 × 100 [Replace with 3.14, r with 50 and h with 100]

\(\approx \) 31400

The lateral surface area of the cylindrical tank is 31400 square inches.

Therefore the cost of painting the tank = $ (31400 × 0.40)

= $12560

Therefore the total cost for Tyler to paint the tank is $12560.

**Question 3: **A can of iced tea is made from a sheet of aluminum that weighs 0.01 ounce per square inch. You receive $0.45 per pound of aluminum that you recycle. How much do you earn for recycling 24 iced tea cans?

**Solution:**

You are given a unit rate in dollars per pound for recycled aluminum, the weight of one square inch of an aluminum can is 0.01 ounce per square inch.

To find the money earned by recycling the 24 cans first find the total surface area of 24 cans using the surface area of each can. Then multiply the per unit rate to find the total money earned.

S = 2\( \pi r^2 \) + 2\(\pi\)rh [Write the formula]

= 2 \(\pi (1.5)^2\) + 2 \(\pi(1.5)(7)\) [Substitute 1.5 for r and 7 for h]

= 4.5\(\pi\) + 21\(\pi\)[Simplify]

= 25.5\(\pi\) [add]

\(\approx \)80 [Use 3.14 for \(\pi\) ]

The surface area of one can is about 80 square inches. So, 24 cans weigh about 24 (80)(0.01) = 19.2 ounces.

Now convert ounce to pound as,

19.2 oz \(~\times~\frac{1~lb}{16~oz} \) = 1.2 oz [1 pound = 16 ounce]

Now find the earning by multiplying the unit cost to the total weight

19.2 \(\times \)$0.45 = $0.54

Therefore, total earnings for recycling 24 cans is $0.54.

**Example 4:** The radius of the cylinder in the image is 7 inches and the height of the cylinder is 14 inches. Find the lateral surface area of the cylinder.

(Take the value of** \(\pi\) ** as 3.14)

**Solution:**

Given,

Radius of the cylinder “r” = 7 inches

Height of the cylinder “h” = 14 inches

Surface area = 2\(\pi\)rh

= 2 × 3.14 × 714 [Replace r with 7 and h with 14]

\(\approx \) 615.44 [Simplify]

The lateral surface area of the cylinder is 615.44 Square inches.

Frequently Asked Questions

The base of the cylinder is a circle. The circumference of the cylinder can be calculated using the formula:

Circumference of cylinder formula = 2\(\pi\)r

Radius of the cylinder is calculated with the help of the cylinder radius formula.

The radius of the cylinder “r” = \(\sqrt{\frac{V}{\pi\times h}}\)

Where V is the volume of the cylinder, h is the height of the cylinder, and is 3.14 or 22/7.

The oblique cylinder is a cylinder that is tilted about its base. The lateral surface of the cylinder is not perpendicular to the base. The circular base of the cylinder does not lie above the other.