What is the difference between lateral surface area and total surface area?

Lateral surface area only includes the curved middle part of the cylinder (the soup can label). Total surface area includes the curved part plus the two circular bases (the top and bottom lids).

What if my cylinder is lying on its side?

The formula does not change. In geometry, the 'height' of a cylinder is always the distance between the two circular bases, even if that distance is horizontal.

Do I always have to multiply by 3.14?

Not always. Many teachers prefer that you leave the symbol $\pi$ in your final answer (like $112\pi$). This is called leaving your answer 'in terms of pi'. Always check the instructions on your assignment!

How do you find the surface area of a cylinder?

To find the surface area of a cylinder, you need to add the area of the two circular bases to the area of the curved side. The formula is $A = 2\pi r^2 + 2\pi rh$ , where $r$ is the radius of the base and $h$ is the height of the cylinder.

Think of a soup can. If you want to know how much metal was used to make the whole can, you have to measure the top lid, the bottom lid, and the paper label that wraps around the middle.

Breaking down the cylinder

A cylinder is made of three distinct parts: a circular top, a circular bottom, and a curved middle section. The top and bottom are called the bases. The curved section is called the lateral surface. To find the total surface area, we simply find the area of these three parts and add them together.

Understanding the formula

The area of one circular base is $\pi r^2$ . Since there are two identical bases (top and bottom), their combined area is $2\pi r^2$ . Now, imagine peeling the label off that soup can and laying it flat. It forms a rectangle. The height of this rectangle is the height of the cylinder ( $h$ ), and the length is the circumference of the base ( $2\pi r$ ). So, the area of the curved part is $2\pi rh$ . Adding them gives our total formula: $A = 2\pi r^2 + 2\pi rh$ .

Where students slip up

The most common mistake is confusing the radius with the diameter. If a problem gives you a diameter of 10, remember to divide it in half to get a radius of 5 before plugging it into the formula. Another common trap is forgetting the '2' in $2\pi r^2$ . If you drop that 2, you are calculating the area of a can missing its top lid!

Worked through

Find the total surface area of a cylinder with a radius of 4 cm and a height of 10 cm. Leave your answer in terms of $\pi$ .

First, identify your given values: $r = 4$ and $h = 10$ .

Next, calculate the area of the two bases. The formula is $2\pi r^2$ . Plug in the radius: $2\pi (4)^2 = 2\pi (16) = 32\pi$ cm $^2$ .

Then, calculate the lateral area. The formula is $2\pi rh$ . Plug in the radius and height: $2\pi (4)(10) = 80\pi$ cm $^2$ .

Finally, add the area of the bases and the lateral area together. $32\pi + 80\pi = 112\pi$ cm $^2$ .

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Where this comes from: OpenStax Pre-Algebra, Chapter 9: Math Models and Geometry · Khan Academy: Volume and surface area unit