Area of Parallelogram
A parallelogram can be cut and rearranged into a rectangle, so its area is \(\text{base} \times \text{perpendicular height}\). This NSW Year 7 maths subtopic (NESA outcome MA4‑ARE‑C‑01) explains why the formula works and why the sloping side is not used.
You will apply \(A = b \times h\), spot the perpendicular height even when it is drawn outside the shape, find a missing base or height from the area, convert units first, and solve problems about car-park spaces, mural panels and mulch for a garden bed. You will also see that a diagonal cuts it into two matching triangles. It is part of Year 7 Area and Volume.
Every question with a fully worked solution.
- Area of Parallelogram - Video - Area of a Parallelogram Watch
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 formula:
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.
| \(4.2 \times 5\) | \(=\) | \(21\) |
\(21~\text{cm}^2\)
| \(24\) | \(=\) | \(b \times 6\) |
| \(b\) | \(=\) | \(24 \div 6 = 4\) |
\(b = 4\) cm
| \(11 \times 5\) | \(=\) | \(55\) |
The \(6\) cm side slopes. The area is \(55~\text{cm}^2\).
| \(3.5 \times 6\) | \(=\) | \(21\) |
The dashed \(6\) cm line meets the base line at a right angle. The area is \(21~\text{cm}^2\).
| \(150 \div 100\) | \(=\) | \(1.5\) |
| \(3.2 \times 1.5\) | \(=\) | \(4.8\) |
| \(4.8 \times 40\) | \(=\) | \(192\) |
Area \(4.8~\text{m}^2\); the paving costs \(\$192\).
(i) Find the height to the \(15\) cm side.
(ii) Find the height to the \(10\) cm side.
| \(120 \div 15\) | \(=\) | \(8\) |
| \(120 \div 10\) | \(=\) | \(12\) |
(i) \(8\) cm
(ii) \(12\) cm. The shorter side has the longer height.
| \(60 \div 100\) | \(=\) | \(0.6\) |
| \(3 \div 0.6\) | \(=\) | \(5\) |
\(h = 5\) m
Common pitfalls
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.