A cylinder is a three-dimensional shape with two congruent circular bases connected by a curved side. Its surface area is the total amount of outside covering on the shape. This matters when you want to paint a can, wrap a label around a container, or find how much material is needed to make a tube with lids.
The key idea is to break the cylinder into simpler flat shapes you already know how to measure.
The two bases are circles, so their combined area is 2πr². The curved side, called the lateral surface, can be unwrapped into a rectangle. The rectangle has height h and width equal to the circumference of the base, 2πr, so its area is 2πrh.
Adding the bases and the lateral area gives SA = 2πr² + 2πrh, which can also be written as SA = 2πr(r + h).
Key Facts
- Total surface area of a cylinder: SA = 2πr² + 2πrh.
- Factored form: SA = 2πr(r + h).
- Area of one circular base: A = πr².
- Area of two circular bases: 2πr².
- Lateral surface area: LSA = 2πrh.
- The unwrapped curved surface is a rectangle with width 2πr and height h.
Vocabulary
- Cylinder
- A three-dimensional solid with two parallel congruent circular bases and one curved surface.
- Radius
- The distance from the center of a circular base to its edge.
- Height
- The perpendicular distance between the two circular bases of a cylinder.
- Lateral surface
- The curved side of a cylinder, not including the two circular bases.
- Surface area
- The total area of all outside faces or surfaces of a three-dimensional object.
Common Mistakes to Avoid
- Using πr²h for surface area. This is the formula for volume, not the amount of outside covering.
- Forgetting one of the circular bases. A closed cylinder has two circles, so the base area must be 2πr².
- Using diameter instead of radius in the formula. If the diameter is given, divide it by 2 before substituting for r.
- Adding 2πr and h for the lateral area. The lateral surface unwraps into a rectangle, so its area is 2πr times h, not a sum.
Practice Questions
- 1 A closed cylinder has radius 4 cm and height 10 cm. Find its total surface area in terms of π, then approximate it using π = 3.14.
- 2 A soup can has diameter 8 cm and height 12 cm. Find the total surface area of the can, using π = 3.14.
- 3 A label wraps around the side of a cylinder but does not cover the top or bottom. Explain which part of the surface area formula should be used and why.