Topic 6 of 12

Area, surface area, and volume

Step 8 · Ratios and negative numbers topics

Find the area of triangles and other shapes by relating them to rectangles, the volume of a box, and the surface area of a box from its faces.

Area

Area is the amount of flat space inside a shape. It is measured in square units, such as square centimeters, written $\text{cm}^2$ . The area of a rectangle is its length times its width:

A = l \times w

Triangles

Cut a rectangle along a diagonal and you get two identical triangles, so each one is half of the rectangle. In the same way, the area of any triangle is half of its base times its height. The height is measured straight up from the base, at a right angle to it.

A = \frac{1}{2} \times b \times h

Example: Area of a triangle

A triangle has a base of 10 cm and a height of 6 cm. Find its area.

Write the formula: $A = \frac{1}{2} \times b \times h$ .
Put in the numbers: $A = \frac{1}{2} \times 10 \times 6$ .
Work it out: $\frac{1}{2} \times 60 = 30$ .

Answer $30 \text{ cm}^2$

Other shapes

For other shapes, cut them into rectangles and triangles, find each area, and add. Sometimes the pieces can be rearranged instead: slice a triangle off one end of a parallelogram, move it to the other end, and you have a rectangle with the same base and height. So the area of a parallelogram is base times height.

Example: A shape made of rectangles

An L-shaped room is made of a 6 m by 4 m rectangle joined to a 3 m by 2 m rectangle. Find its area.

Area of the first rectangle: $6 \times 4 = 24$ .
Area of the second rectangle: $3 \times 2 = 6$ .
Add them: $24 + 6 = 30$ .

Answer $30 \text{ m}^2$

Volume of a box

Volume is the amount of space inside a solid. It is measured in cubic units, such as $\text{cm}^3$ . A box, also called a rectangular prism, with length $l$ , width $w$ and height $h$ holds $l \times w \times h$ unit cubes:

V = l \times w \times h

The formula works for fractions and decimals too. A box 4 cm long, $2\frac{1}{2}$ cm wide and 3 cm high has volume $4 \times 2\frac{1}{2} \times 3 = 30 \text{ cm}^3$ .

Surface area

Surface area is the total area of all the outside faces of a solid. Imagine unfolding a box so it lies flat. The flat pattern is called a net. A box has 6 rectangular faces in 3 matching pairs: top and bottom, front and back, left and right. Find the area of one face from each pair, double it, and add.

S = 2lw + 2lh + 2wh

Example: Surface area of a box

Find the surface area of a box 5 cm long, 3 cm wide and 2 cm high.

Top and bottom: $2 \times 5 \times 3 = 30$ .
Front and back: $2 \times 5 \times 2 = 20$ .
Left and right: $2 \times 3 \times 2 = 12$ .
Add the three pairs: $30 + 20 + 12 = 62$ .

Answer $62 \text{ cm}^2$

Try it yourself

Work each one on paper first, then open the answer to check.

Find the area of a triangle with base 8 m and height 5 m.
Show answer
$20 \text{ m}^2$ . $\frac{1}{2} \times 8 \times 5 = 20$ .
Find the area of a parallelogram with base 9 cm and height 4 cm.
Show answer
$36 \text{ cm}^2$ . $9 \times 4 = 36$ .
Find the volume of a box 6 cm by 4 cm by 5 cm.
Show answer
$120 \text{ cm}^3$ . $6 \times 4 \times 5 = 120$ .
Find the surface area of a cube with edges 3 cm long.
Show answer
$54 \text{ cm}^2$ . Each face is $3 \times 3 = 9$ , and there are 6 faces: $6 \times 9 = 54$ .
A fish tank is 50 cm long, 30 cm wide and 40 cm high. What is its volume?
Show answer
$60{,}000 \text{ cm}^3$ . $50 \times 30 \times 40 = 60{,}000$ .
A box is $1\frac{1}{2}$ m long, 2 m wide and $\frac{1}{2}$ m high. Find its volume.
Show answer
$1\frac{1}{2} \text{ m}^3$ . $\frac{3}{2} \times 2 \times \frac{1}{2} = \frac{3}{2}$ .