Math Core

How much wrapping paper covers a gift box? How much paint covers a wooden block? These questions are not about the space inside a solid. They are about the flat faces on the outside. To see all those faces at once, you can unfold the solid into a flat pattern called a net.

Nets

Definition

Net

A net is a flat pattern of shapes that folds up to make a three-dimensional solid. Every face of the solid appears in the net exactly once.

A rectangular prism (a box) has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right. Here is one net of a box that is 4 inches long, 3 inches wide and 2 inches tall.

A net of a box 4 in long, 3 in wide and 2 in tall. The middle column folds up into the bottom, front, top and back. The two small rectangles fold up into the sides.

Other solids have nets too. A square pyramid has a square base and 4 triangles that meet at a point. A triangular prism has 2 triangles (its ends) and 3 rectangles.

Surface area

Definition

Surface area

The surface area of a solid is the total area of all its faces. It is measured in square units, because it is a flat area, even though the solid is 3D.

To find surface area, find the area of each face in the net and add them all.

Worked example: Surface area of a box

Find the surface area of the box whose net is shown above.

The net has three pairs of matching rectangles:

Top and bottom: $4 \times 3 = 12$ each, so $2 \times 12 = 24$ .
Front and back: $4 \times 2 = 8$ each, so $2 \times 8 = 16$ .
Two sides: $2 \times 3 = 6$ each, so $2 \times 6 = 12$ .

Total: $24 + 16 + 12 = 52$ . The surface area is 52 square inches.

Finding surface area

Draw or picture the net, find the area of every face, and add. For a box, the faces come in 3 pairs, so you can find one face from each pair, add them, and double.

A cube is a box with all edges the same length $s$ . Its 6 faces are identical squares, so its surface area is $6 \times s \times s$ .

Worked example: A cube

A cube has edges of 3 cm. Find its surface area.

Each face: $3 \times 3 = 9$ square cm. Six faces: $6 \times 9 = 54$ square cm.

Pyramids and triangular prisms

A net of a square pyramid. The square base has 6 cm sides. Each triangle has a base of 6 cm and a height of 4 cm (the vertical segment in the top triangle).

Worked example: A square pyramid

Find the surface area of the pyramid whose net is shown above.

Square base: $6 \times 6 = 36$ square cm.
Each triangle: $\dfrac{1}{2} \times 6 \times 4 = 12$ square cm. There are 4 of them: $4 \times 12 = 48$ square cm.

Total: $36 + 48 = 84$ square centimeters.

Worked example: A triangular prism

A tent is shaped like a triangular prism. Each end is a right triangle with legs of 3 ft and 4 ft and a slanted side of 5 ft. The tent is 10 ft long. What is its surface area, including the floor?

The net has 2 triangles and 3 rectangles.

Two triangles: each is $\dfrac{1}{2} \times 3 \times 4 = 6$ , so $2 \times 6 = 12$ square feet.
Three rectangles, all 10 ft long: $3 \times 10 = 30$ , $4 \times 10 = 40$ and $5 \times 10 = 50$ , for $120$ square feet.

Total: $12 + 120 = 132$ square feet.

Common mistake

Don't mix up surface area and volume. Surface area covers the outside: it adds the areas of the faces and uses square units. Volume fills the inside: it multiplies length, width and height and uses cubic units. Wrapping and painting are surface area; filling and packing are volume.

Tip

Before adding, count your faces. A box needs 6, a square pyramid needs 5, and a triangular prism needs 5. If your list is short a face, your answer will be too small.

Practice

Practice 1

A cube has edges of 5 inches. What is its surface area, in square inches?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

A box is 6 cm long, 4 cm wide and 3 cm tall. What is its surface area, in square centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

A net is made of one square and four identical triangles. What solid does it fold into?

Practice 4

Mara wants to know how much paint she needs to cover every side of a wooden block. What should she find?

Practice 5

A square pyramid has a base with 8-inch sides. Each triangular face has a base of 8 inches and a height of 5 inches. What is the surface area, in square inches?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A cube has edges of $2\frac{1}{2}$ inches. What is its surface area, in square inches?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A box with no lid has a base that is 10 cm by 6 cm and a height of 4 cm. How much cardboard is needed to make it, in square centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A triangular prism is 9 inches long. Each triangular end has a base of 6 inches, a height of 4 inches, and two slanted sides of 5 inches. What is the surface area of the prism, in square inches?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.