Math Made Easy! How to Find the Surface Area of a Cylinder

Total Surface Area of a Cylinder

For high school geometry students that are not really "fans" of the geometry subject, it is problems like finding the surface area of a cylinder that often cause kids to shut their text books and give up or find a geometry tutor.

But, don't panic just yet. Geometry, like many types of math, is often so much easier to understand when broken down into bite-sized pieces. This geometry tutorial will do just that - break down the equation for finding the surface area of a cylinder into easy to understand portions.

Be sure to follow along the cylinder surface area problems and solutions in the Geometry Help Online section below, as well as to try out the Math Made Easy! quiz.

S.A. = 2πr² + 2πrh

Where: r is the radius of the cylinder and h is height of the cylinder.

Before beginning be sure you understand the following geometry tutorials:

• how to find the area of a circle
• how to find the circumference of a circle

Admittedly, the formula for the surface area of a cylinder isn't too pretty. So, let's try to break the formula apart into understandable pieces. A good math tip is to try to visualize the geometrical shape with an object with which you are already familiar.

What objects in your home are cylinders? I know in my pantry I have a lot of cylinders - better known as canned goods.

Let's examine a can. A can is made up of a top and bottom and a side that curves around. If you could unfold the side of a can it would actually be a rectangle. While I am not going to unfold a can, I can easily unfold the label around it and see that it is a rectangle.

Now you can visualize the total surface area of a cylinder (can):

• a can has 2 circles, and
• a can has 1 rectangle

In other words, you can think of the equation of the total area of a cylinder as:

S.A. = (2)(area of a circle) + (area of a rectangle)

Therefore, in order to calculate the surface area of a cylinder you need to calcuate the area of a circle (twice) and the area of a rectangle (once).

Let's look at the total surface area of a cylinder equation again and break it down into easy to understand portions.

Area of Cylinder = 2πr²(portion 1) + 2πrh (portion 2)

• Portion 1: The first portion of the cylinder equation has to do with the area of the 2 circles (the top and bottom of the can). Since we know that the area of one circle is πr² then the area of two circles is 2πr² . So, the first part of the cylinder equation gives us the area of the two circles.
• Portion 2: The second portion of the equation gives us the area of the rectangle that curves around the can (the unfolded label in our canned good example).We know that the area of a rectangle is simply its width (w) times its height (h). So why is the width in the second portion of the equation (2πr)(h) written as (2πr)? Again, picture the label. Notice that the width of the rectangle when rolled back around the can is exactly the same thing as the circumference of the can. And the equation for circumference is 2πr. Multiply (2πr) times (h) and you have the area of the rectangle portion of the cylinder.

Check out three common types of geometry problems for finding the surface area of a cylinder given various measurements.

Problem: Find the total surface area of a cylinder with a radius of 5 cm. and a height of 12 cm.

Solution: Since we know r = 5 and h=12 substitute 5 in for r and 12 in for h in the cylinder's surface area equation and solve.

• S.A. = (2)π(5)² + (2)π(5)(12)
• S.A. = (2)(3.14)(25) + (2)(3.14)(5)(12)
• S.A. = 157 + 376.8
• S.A. = 533.8

Answer: The surface area of a cylinder with a radius of 5 cm. and a height of 12 cm. is 533.8 cm. squared.

Problem: What is the total surface area of a cylinder with a diameter of 4 in. and a height of 10 in.?

Solution: Since the diameter is 4 in., we know that the radius is 2 in., since the radius is always 1/2 of the diameter. Plug in 2 for r and 10 for h in the equation for the surface area of a cylinder and solve:

• S.A. = 2π(2)² + 2π(2)(10)
• S.A. = (2)(3.14)(4) + (2)(3.14)(2)(10)
• S.A. = 25.12 + 125.6
• S.A. = 150.72

Answer: The surface area of a cylinder with a diameter of 4 in. and a height of 10 in. is 150.72 in. squared.

Problem: The area of one end of a cylinder is 28.26 sq. ft. and its height is 10 ft. What is the total surface area of the cylinder?

Solution: We know that the area of a circle is πr² and we know that in our example the area of one end of the cylinder (which is a circle) is 28.26 sq. ft. Therefore, substitute 28.26 for πr² in the formula for the area of a cylinder. You can also substitute 10 for h since that is given.

S.A. = (2)(28.26) + 2πr(10)

This problem still cannot be solved since we do not know the radius, r. In order to solve for r we can use the area of a circle equation. We know that the area of the circle in this problem is 28.26 ft. so we can substitute that in for A in the area of a circle formula and then solve for r:

• Area of Circle (solve for r):
• 28.26 = πr²
• 9 = r²(divide both sides of the equation by 3.14)
• r = 3 (take the square root of both sides of the equation)

Now that we know r = 3 we can substitute that into the area of the cylinder formula along with the other substitutions, as follows:

• S.A. = (2)(28.26) + 2π(3)(10)
• S.A. = (2)(28.26) + (2)(3.14)(3)(10)
• S.A. = 56.52 + 188.4
• S.A. = 244.92

Answer: The total surface area of a cylinder whose end has an area of 28.26 sq. ft. and a height of 10 is 244.92 sq. ft.

If you have another specific problem you need help with related to the total surface area of the cylinder please ask in the comment section below. I'll be glad to help out and may even include your problem in the problem/solution section above.

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