Area of a Parallelogram –

The parallelogram is a shape that trips up many GCSE students because it looks like a slanted rectangle — and the temptation is to multiply the two visible side lengths. The correct formula uses the perpendicular height, not the slant side.

What Is a Parallelogram?

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal in length. Rectangles and rhombuses are special cases of parallelograms, but in GCSE questions a "parallelogram" usually means the slanted version where the angles are not 90°.

The area formula works because you can "cut" a right-angled triangle from one end and move it to the other end, turning the parallelogram into a rectangle with the same base and perpendicular height.

This is why A = base × perpendicular height gives the same result as the area of that equivalent rectangle. The slant side is irrelevant to the area — it only affects the perimeter.

Diagrams often show a dashed line from one vertex perpendicular to the base — this dashed line is the height you need.

Key Formulas

Step-by-Step Method

1. Identify the base — either pair of parallel sides can serve as the base.
2. Find the perpendicular height — the distance measured at 90° between the base and the opposite side.
3. Multiply: A = base × perpendicular height.

Worked Example 1 — Foundation Level

Question: A parallelogram has a base of 8 cm and a perpendicular height of 5 cm. Find its area.

Working: A = base × height A = 8 × 5 A = 40

Answer: 40 cm²

Worked Example 2 — Higher Level

Question: A parallelogram has an area of 91 cm² and a base of 13 cm. Calculate the perpendicular height.

Working: A = base × height 91 = 13 × h h = 91 ÷ 13 h = 7

Answer: 7 cm

Worked Example 3 — Exam Style

Question: A compound shape consists of a parallelogram on top of a rectangle. The rectangle is 10 cm by 4 cm. The parallelogram shares the 10 cm side as its base and has a perpendicular height of 3 cm. Find the total area.

Working: Rectangle area = 10 × 4 = 40 cm² Parallelogram area = 10 × 3 = 30 cm² Total area = 40 + 30 = 70

Answer: 70 cm²

Common Mistakes

• Using the slant height instead of the perpendicular height. The slant edge is a side of the shape, not the height. The perpendicular height is always shorter than the slant side and is marked with a right-angle symbol.
• Confusing with a trapezium. A parallelogram has two pairs of parallel sides, not one. You do not need to average the sides — both parallel sides are equal, so just use one as the base.
• Halving the answer. Unlike a triangle, there is no ½ in the parallelogram formula. If you halve, you have calculated the area of a triangle instead.
• Forgetting square units. Always state your answer in cm², m², or the appropriate square unit.
• Multiplying two adjacent sides. Two adjacent sides of a parallelogram are not the same as base and height unless the shape is a rectangle.

Exam Tips

• The parallelogram formula is not always on the formula sheet — learn it by heart.
• If a right-angle symbol appears inside the shape, that line is the perpendicular height.
• When the perpendicular height is outside the shape (drawn as an extension), it is still valid — this happens with very slanted parallelograms.
• If you are given the side lengths and an angle instead of the height, you can use h = side × sin(angle) to find the perpendicular height before applying the formula.
• Sketch the equivalent rectangle alongside the parallelogram if you find it helpful to visualise the area.

Practice Questions

Q1 (Foundation): A parallelogram has a base of 12 cm and a perpendicular height of 7 cm. Find its area.

Q2 (Foundation): A parallelogram has a base of 9.5 cm and a perpendicular height of 4 cm. Find its area.

Q3 (Higher): A parallelogram has an area of 108 cm². The perpendicular height is 9 cm. The slant side is 11 cm. Find the base.

Practise area of a parallelogram questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Area of 2D Shapes
• Area of a Trapezium
• Perimeter

Summary

• A parallelogram has two pairs of parallel and equal sides.
• Its area is A = base × perpendicular height — do not use the slant side.
• The formula has no ½ — unlike a triangle.
• The perpendicular height is at right angles to the base, often marked with a small square symbol.
• In compound-shape questions, calculate the parallelogram area separately before combining.
• A rectangle is a special case of a parallelogram where the perpendicular height equals the side length.