Area of rectangles, triangles and parallelograms | Year 7

01 / LEARN

Understand the idea.

Area counts square units

Area measures how many square units cover a surface. A rectangle contains equal rows of unit squares, giving length times width. Cutting a triangle off one end of a parallelogram and sliding it to the other end makes a rectangle of equal area.

The parallelogram formula uses base times perpendicular height. The height measures the shortest distance between its parallel sides, meeting the base at a right angle. A sloping side is not generally that height.

\begin{aligned} A_{rectangle} & = l w \\ A_{parallelogram} & = b h \end{aligned}

A triangle is half a parallelogram

Two matching copies of a triangle form a parallelogram with the same base and perpendicular height. One triangle therefore occupies half of $b h$ . Choose any base, but use the height perpendicular to that particular base or its extension.

For a composite region, split the surface into familiar shapes without overlap, or subtract a missing piece from a larger shape. Label the base-height pairs and use one length unit throughout. The final area uses square units.

A_{triangle} = \frac{1}{2} b h

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Triangle height, not the sloping side

A triangle has base 12 cm and perpendicular height 5 cm. Find its area.

Worked solution

1
Choose the base and its perpendicular height.
$b = 12 cm, h = 5 cm$
The question supplies the required right-angle height.
2
Substitute into the triangle area formula.
$A = \frac{1}{2} b h = \frac{1}{2} \times 12 \times 5$
A triangle occupies half the matching parallelogram.
3
Calculate and include square units.
$A = 6 \times 5 = 30 {cm}^{2}$
Multiplying two lengths produces an area.

Answer

$30 {cm}^{2}$

EXAMPLE 02

Parallelogram with extra information

A parallelogram has base 10 cm, perpendicular height 6 cm and sloping side 7 cm. Find its area.

Worked solution

1
Select the perpendicular height.
$b = 10 cm, h = 6 cm$
The seven-centimetre sloping side is not perpendicular to the base.
2
Use the parallelogram area relationship.
$A = b h = 10 \times 6$
Rearranging a parallelogram gives a rectangle with this base and height.
3
State the area.
$A = 60 {cm}^{2}$
The sloping side length is not needed for this calculation.

Answer

$60 {cm}^{2}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the area of a triangle with base 8 m and perpendicular height 3 m.

Show answer & reasoning

12 m²

The triangle is half a parallelogram of area $8 \times 3 = 24 m^{2}$ . Its area is $\frac{1}{2} \times 24 = 12 m^{2}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Find the area of a rectangle 9 cm by 6 cm.
1 mark
Find the area of a triangle with base 14 cm and perpendicular height 5 cm.
2 marks
A parallelogram has base 11 m, perpendicular height 4 m and sloping side 5 m. Find its area.
2 marks
A triangle has area 36 cm² and base 12 cm. Find its perpendicular height.
3 marks
Find the area of the L-shaped region shown.
3 marks
A triangle and a parallelogram have the same base and perpendicular height. The parallelogram has area 48 cm². Find the triangle area and explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the area of a rectangle 9 cm by 6 cm.
1 mark
$9 \times 6 = 54$ cm².
54 cm²
Find the area of a triangle with base 14 cm and perpendicular height 5 cm.
2 marks
$\frac{1}{2} \times 14 \times 5 = 35$ cm².
35 cm²
A parallelogram has base 11 m, perpendicular height 4 m and sloping side 5 m. Find its area.
2 marks
Use height 4 m, not sloping side 5 m.
$11 \times 4 = 44$ m².
44 m²
A triangle has area 36 cm² and base 12 cm. Find its perpendicular height.
3 marks
$36 = \frac{1}{2} \times 12 \times height$ .
$36 = 6 \times height$ .
$Height = 36 \div 6 = 6$ cm.
6 cm
Find the area of the L-shaped region shown.
3 marks
Full rectangle: $9 \times 7 = 63$ m².
Missing rectangle: $4 \times 3 = 12$ m².
$63 - 12 = 51$ m².
51 m²
A triangle and a parallelogram have the same base and perpendicular height. The parallelogram has area 48 cm². Find the triangle area and explain.
2 marks
A triangle has half the area for the same base and height.
$48 \div 2 = 24$ cm².
24 cm²

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 7 measurement and spatial representation: areas of triangles/parallelograms, prism volumes, circle circumference, nets and views. Precision is integrated into practical measurement; circle area is not introduced here.

Official curriculum source ↗

In this lesson

Understand the idea.

Area counts square units

A triangle is half a parallelogram

See the method in action.

Triangle height, not the sloping side

Choose the base and its perpendicular height.

Substitute into the triangle area formula.

Calculate and include square units.

Parallelogram with extra information

Select the perpendicular height.

Use the parallelogram area relationship.

State the area.

Try it yourself.

Check your understanding

Keep practising