Year 7 Maths Area and Volume

Area of Parallelogram

Q: What is the formula for the area of a parallelogram?

A = b × h: base times perpendicular height.

Q: How do you find the height of a parallelogram from its area?

Divide the area by the base. 120 ÷ 15 = 8.

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Theory

A parallelogram is a four-sided shape with two pairs of parallel sides. Cut a triangle off one end and slide it to the other end: you get a rectangle, so $A = b \times h$ (base times perpendicular height, not the slant height). This Year 7 maths page shows the formula, sloping sides, heights outside the shape and missing lengths.

Cut the right-angled triangle off one end of the parallelogram and slide it to the other end. Nothing is lost, so the area does not change. The new shape is a rectangle $b$ long and $h$ high, so $A = b \times h$ (or $A = bh$).

$b$ is the base length. $h$ is the perpendicular height: the distance straight across between the base and the opposite side, meeting the base at a right angle. The sloping sides (the slant height) are not used.

The height can be outside the shape. If the parallelogram leans a long way, the height meets the base line made longer. It is still the height.

Any side can be the base. Each base has its own height, but the area is the same. Missing base or height: divide the area by the length you know.

A diagonal cuts a parallelogram into two equal triangles, so each triangle is $\dfrac{1}{2} \times b \times h$. Change the lengths to the same unit before you multiply.

The red triangle slides to the dashed place on the right and makes a rectangle $b$ long and $h$ high: $A = b \times h$.

The height is outside the shape, meeting the base line made longer. $A = b \times h$ still works.

The formula:

\[ A = b \times h \]

A = b \times h

\[ h = A \div b, \qquad b = A \div h \]

h = A \div b, b = A \div h

          A sloping side is not the height. Base \(9\) cm, sloping side \(7\) cm, height \(5\) cm: \(A = 9 \times 5 = 45~\text{cm}^2\), not \(9 \times 7\).
        

          Don’t halve. The half is for a triangle. A parallelogram is a whole rectangle’s worth: \(A = b \times h\).
        

How to find the area of a parallelogram

Find the base and the perpendicular height (ignore the sloping sides).
Change them to the same unit.
Multiply: $A = b \times h$, in square units.

Example 1 — Base and height

Find the area of a parallelogram with base length $4.2$ cm and perpendicular height $5$ cm.

Solution

$4.2 \times 5$

$=$

$21$

4.2 \times 5 = 21

$21~\text{cm}^2$

Example 2 — Finding the base

The area of a parallelogram is $24~\text{cm}^2$ and its height is $6$ cm. Find the base length.

Solution

$24$	$=$	$b \times 6$
$b$	$=$	$24 \div 6 = 4$

24 = b \times 6, b = 24 \div 6 = 4

$b = 4$ cm

Example 3 — Ignore the sloping side

Find the area of the parallelogram.

Solution

$11 \times 5$

$=$

$55$

11 \times 5 = 55

The $6$ cm side slopes. The area is $55~\text{cm}^2$.

Example 4 — Height outside

This parallelogram leans a long way. Find its area.

Solution

$3.5 \times 6$

$=$

$21$

3.5 \times 6 = 21

The dashed $6$ cm line meets the base line at a right angle. The area is $21~\text{cm}^2$.

Example 5 — Units and cost

A paved area is a parallelogram with a base of $3.2$ m and a perpendicular height of $150$ cm. Paving costs $\$40$ per square metre. Find the cost.

Solution

$150 \div 100$	$=$	$1.5$
$3.2 \times 1.5$	$=$	$4.8$
$4.8 \times 40$	$=$	$192$

150 cm = 1.5 m, 3.2 \times 1.5 = 4.8, 4.8 \times 40 = 192

Area $4.8~\text{m}^2$; the paving costs $\$192$.

Example 6 — One area, two heights

A parallelogram has sides $15$ cm and $10$ cm. Its area is $120~\text{cm}^2$.
(i) Find the height to the $15$ cm side.
(ii) Find the height to the $10$ cm side.

Solution

$120 \div 15$	$=$	$8$
$120 \div 10$	$=$	$12$

120 \div 15 = 8, 120 \div 10 = 12

(i) $8$ cm
(ii) $12$ cm. The shorter side has the longer height.

Example 7 — Area in m², length in cm

A sign shaped like a parallelogram has an area of $3~\text{m}^2$ and a base of $60$ cm. Find its height.

Solution

$60 \div 100$	$=$	$0.6$
$3 \div 0.6$	$=$	$5$

60 cm = 0.6 m, 3 \div 0.6 = 5

$h = 5$ m

Common pitfalls

Using the slant height. Multiply by the perpendicular height, not the sloping side.

Halving. $A = b \times h$ for a parallelogram; only a triangle is halved.

Mixed units. Base $45$ mm and height $3$ cm: $4.5 \times 3 = 13.5~\text{cm}^2$.

Wrong height for the base. Each base has its own height, at a right angle to that base.

Frequently asked questions

What is the formula for the area of a parallelogram?

$A = b \times h$: base times perpendicular height.

Why does a parallelogram have the same area as a rectangle?

Cut a triangle off one end and slide it to the other end to make a rectangle with the same base and height.

Do you use the sloping side?

No. Use the perpendicular height.

How do you find the height of a parallelogram from its area?

Divide the area by the base. $120 \div 15 = 8$.

What is the area of a triangle cut off by a diagonal?

Half the parallelogram.

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Area of Triangle

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Area of Composite Shapes

\(24\)	\(=\)	\(b \times 6\)
\(b\)	\(=\)	\(24 \div 6 = 4\)

\(150 \div 100\)	\(=\)	\(1.5\)
\(3.2 \times 1.5\)	\(=\)	\(4.8\)
\(4.8 \times 40\)	\(=\)	\(192\)

\(120 \div 15\)	\(=\)	\(8\)
\(120 \div 10\)	\(=\)	\(12\)

\(60 \div 100\)	\(=\)	\(0.6\)
\(3 \div 0.6\)	\(=\)	\(5\)

Area of Parallelogram

📖 Theory

How to find the area of a parallelogram

Common pitfalls

Frequently asked questions

Theory