Lateral vs total surface area
Lateral and Total Surface Area
Lateral and Total Surface Area is a topic in Three-Dimensional Measurement in the Common Core State Standards. It is aligned to Standard G-MG.1, which requires students to distinguish and compute the lateral and total surface area of solids to model objects.
Lateral surface area covers only the sides of a solid, while total surface area adds the base or bases.
Theory
Surface area comes in two forms:
- Lateral surface area — the side surfaces only, excluding the base(s).
- Total surface area — the lateral area plus the base(s).
For a cylinder the lateral surface unrolls into a rectangle of width \(2\pi r\) (the circumference) and height \(h\), giving lateral area \(2\pi rh\). For a prism, lateral area is perimeter times height.
Lateral areas:
How to find lateral and total area
- Lateral: the side surface — \(2\pi rh\) or perimeter \(\times h\).
- Add the base(s) for total surface area.
- Read the problem to see if bases are included (open vs closed).
The lateral surface is the curved side only.
| \(\text{lateral}\) | \(=\) | \(2\pi rh=2\pi(3)(5)=30\pi\) |
Add the two circular bases to the lateral area.
| \(\text{total}\) | \(=\) | \(30\pi+2\pi(3)^2\) |
| \(=\) | \(30\pi+18\pi=48\pi\) |
Subtract lateral from total.
| \(48\pi-30\pi\) | \(=\) | \(18\pi\) |
Lateral area of a prism is perimeter times height.
| \(\text{lateral}\) | \(=\) | \(20\times 7=140\) |
Common pitfalls
Frequently asked questions
What is lateral surface area?
The area of the side surfaces only, not including the base or bases.
What is total surface area?
The lateral surface area plus the area of the base(s).
What is the lateral area of a cylinder?
\(2\pi rh\) — the curved side unrolled into a rectangle.
How do you find a prism's lateral area?
Multiply the base perimeter by the height.