Surface Area of Prisms & Cylinders

Q: What is lateral area versus total surface area?

Lateral area is just the wall around the sides ($P \cdot h$), leaving out the top and bottom. Total surface area adds the two bases back in.

Surface area is the total area of every face of a solid — the amount of wrapping paper it would take to cover it with no overlap. For prisms and cylinders it splits neatly into two bases plus the wall that wraps around them.

The wall is called the lateral area, and it is always the perimeter of the base times the height. Add the two bases and you have the full surface area. One idea covers every prism and the cylinder too.

The two-part formula

Total surface area = lateral area + 2 × (base area). The lateral area is $P \cdot h$ , the base perimeter times the height. For a rectangular prism that is $2(lw + lh + wh)$ once you expand it out.

For a cylinder the base is a circle, so the perimeter becomes the circumference $2\pi r$ and the two bases add $2\pi r^2$ . That gives surface area $= 2\pi r h + 2\pi r^2$ . The figure marks the radius $r$ of the circular base and the height $h$ that the wall wraps.

Keeping the pieces straight

The most common mistake is forgetting one of the two bases, or doubling the lateral area by accident. Write the two pieces separately, then add.

Units are always squared — square inches, square centimeters — because area measures a surface, not a length or a volume.

Worked examples

Example 1: a rectangular prism

Find the surface area of a $5$ by $4$ by $3$ rectangular prism.

Use the prism formula

2(lw + lh + wh)

Substitute

2(5\cdot4 + 5\cdot3 + 4\cdot3)

Add inside, then double

2(20 + 15 + 12) = 94

Answer: $94$ square units

Example 2: a cylinder

Find the surface area of a cylinder with radius $3$ and height $5$ .

Lateral area plus two bases

2\pi r h + 2\pi r^2

Substitute

2\pi(3)(5) + 2\pi(3)^2

Simplify

30\pi + 18\pi = 48\pi

Answer: $48\pi$ square units

Try one yourself

Common questions

What is lateral area versus total surface area?

Lateral area is just the wall around the sides ( $P \cdot h$ ), leaving out the top and bottom. Total surface area adds the two bases back in.

Why is the cylinder's wall

2\pi r h

Unroll the wall and it is a rectangle: its height is $h$ and its width is the circle's circumference $2\pi r$ . Area of that rectangle is $2\pi r \cdot h$ .

Do I leave answers in terms of pi?

Usually yes for exact answers. If a decimal is asked for, multiply by $3.14159$ at the very end so rounding does not build up.

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Surface Area of Prisms & Cylinders

The two-part formula

Keeping the pieces straight

Worked examples

Example 1: a rectangular prism

Example 2: a cylinder

Try one yourself

Common questions

Keep learning

Surface Area of Pyramids & Cones

Volume of Pyramids & Cones

Solids, Nets & Cross-Sections