Surface Area of Prisms & Cylinders

Surface area is how much material it takes to cover the outside of a solid. Think about the cardboard in a cereal box or the paper label wrapped around a soup can. You are not filling the solid up, which is volume. You are covering it, so every answer comes out in square units.

The clearest way to see it is to flatten the solid out. Cut a box along its edges and it opens into six flat rectangles. Peel the label off a can and it unrolls into one long rectangle. That flat picture is called a net, and once you have it, surface area is just the area of some flat shapes added together.

Lateral means the sides only

A prism or a cylinder has two identical ends, called the bases, and a wall that wraps around between them. Lateral surface area is that wall by itself, with no top and no bottom. Total surface area is the wall plus both bases. Problems ask for one or the other by name, so read for the word lateral before you start.

The wall is one rectangle in disguise. Unroll it and its height is the height of the solid, while its width is the distance all the way around the base. That gives the lateral formula $L = Ph$ , where $P$ is the perimeter of the base and $h$ is the height. Add the two bases and you have $S = Ph + 2B$ , with $B$ standing for the area of one base.

Cylinders use the same two formulas

A cylinder is the same idea with a circle for a base. The distance around a circle is its circumference $2\pi r$ , so the unrolled wall is a rectangle measuring $2\pi r$ across and $h$ tall. That makes the lateral surface area $L = 2\pi r h$ , which is exactly $Ph$ with the circle's perimeter substituted in.

Each circular base has area $\pi r^2$ , and there are two of them, so the total is $S = 2\pi r h + 2\pi r^2$ . Compute the two pieces separately and write them on separate lines before you add. If the figure marks the diameter instead of the radius, halve it first, because both formulas run on $r$ .

Habits that keep the answer clean

Label the base before you compute anything. On a can it is obviously the circle, but a rectangular prism can rest on any face, and the problem decides which pair counts as the bases by how it shades or describes them. Once the base is fixed, $P$ and $B$ come from that shape and $h$ is measured perpendicular to it.

The two mistakes to guard against are dropping a base and forgetting the $2$ in front of it. A cube is the friendly reminder: six identical square faces, so its surface area is $6s^2$ , and every one of the six has to be counted. Give every answer in square units, and if a decimal is asked for, substitute $3.14$ for $\pi$ only at the last step.

Worked examples

Example 1: a rectangular prism

A prism has a $3$ by $2$ base and a height of $4$ . Find its lateral and total surface area.

Perimeter of the base

P = 3 + 2 + 3 + 2 = 10

Lateral area wraps the sides

L = Ph = 10 \cdot 4 = 40

Area of one base

B = 3 \cdot 2 = 6

Add both bases

S = 40 + 2 \cdot 6 = 52

Answer: $L = 40$ square units and $S = 52$ square units.

Example 2: a cylinder

A cylinder has a radius of $2$ and a height of $5$ . Find its lateral and total surface area. Use $3.14$ for $\pi$ .

Lateral area is the unrolled wall

L = 2\pi r h = 2 \cdot 3.14 \cdot 2 \cdot 5 = 62.8

Both circular bases

2\pi r^2 = 2 \cdot 3.14 \cdot 2^2 = 25.12

Add the wall and the bases

S = 62.8 + 25.12 = 87.92

Answer: $L = 62.8$ square units and $S = 87.92$ square units.

Example 3: a cube

Find the total surface area of a cube with a side length of $3$ .

A cube has

6

identical square faces

S = 6s^2

Substitute the side length

S = 6 \cdot 3^2

Square first, then multiply

S = 6 \cdot 9 = 54

Answer: $54$ square units.

Example 4: wrapping a gift box

A gift box has a $6$ inch by $4$ inch base and a height of $5$ inches. How much paper is needed to cover the whole box, with no overlap?

Covering the whole box means total surface area

S = Ph + 2B

Perimeter of the base

P = 6 + 4 + 6 + 4 = 20

Wrap the sides

Ph = 20 \cdot 5 = 100

Area of one base, then add both

B = 24,\; S = 100 + 2 \cdot 24 = 148

Answer: $148$ square inches of paper.

Try one yourself

Common questions

What is the difference between lateral and total surface area?

Lateral surface area is the wall around the sides only, $L = Ph$ . Total surface area adds the two bases back on, $S = Ph + 2B$ . A soup can label is lateral area; the label plus both metal ends is total surface area.

How is surface area different from volume?

Surface area covers the outside of a solid and comes out in square units, like square inches. Volume fills the inside and comes out in cubic units. Wrapping paper is surface area; the cereal inside the box is volume.

Why is a cylinder's wall

2\pi r h

Unroll the wall and it flattens into a rectangle. Its height is $h$ and its width is the distance around the circular base, which is the circumference $2\pi r$ . Multiply the two sides of that rectangle and you get $2\pi r h$ .

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Surface Area of Prisms & Cylinders

Lateral means the sides only

Cylinders use the same two formulas

Habits that keep the answer clean

Worked examples

Example 1: a rectangular prism

Example 2: a cylinder

Example 3: a cube

Example 4: wrapping a gift box

Try one yourself

Common questions

Keep learning

Volume of Cylinders

Surface Area of Prisms & Cylinders

Missing Dimensions & Composite Solids