Area formulas, made visual
Drag the sliders to resize any 2D shape and watch the area formula update in real time. Built for Year 4–7 maths and scholarship preparation — rectangle, triangle, circle, parallelogram, trapezium, rhombus, and regular polygons.
Calculate the area of a rectangle
A = 40 cm²
Tip: drag any slider and watch the formula recompute. Try the Eye button to hide the formula and challenge yourself.
How to find the area of any 2D shape
Every shape has a formula, but the trick isn't memorising seven of them — it's recognising what makes each one work. This guide walks through the area formulas your child needs for Year 4–7 maths and scholarship exams, with a live diagram you can drag above to test each one.
Five steps that work for any shape
When you face an area question — even an unfamiliar shape — these five steps always apply: identify the shape, pick the right formula, find the dimensions, substitute, then calculate and label the units. The interactive tool above is a sandbox for practising step 4 (substitution) until it's second nature.
Area of a rectangle
The area of a rectangle is the length multiplied by the width. Both dimensions must be in the same unit, and the answer is always in square units (cm², m², mm²).
Example: A 6 cm × 4 cm rectangle has an area of 6 × 4 = 24 cm².
This is the cornerstone formula. Every other area formula either reduces to it (squares, parallelograms) or builds from it (triangles are half of a rectangle).
Area of a square
A square is a special rectangle where every side is the same length, so the formula collapses to side squared.
Example: A square with 7 cm sides has an area of 7² = 49 cm².
When the side is a whole number, the area always lands on a perfect square (1, 4, 9, 16, 25 …) — which is where the word 'squared' comes from.
Area of a parallelogram
A parallelogram has two pairs of parallel sides. Its area is the base multiplied by the perpendicular height. The height must be the perpendicular distance between the parallel sides — not the slanted edge.
Example: A parallelogram with a 7 cm base and a 4 cm perpendicular height has an area of 7 × 4 = 28 cm².
Think of a parallelogram as a rectangle pushed sideways. The area stays identical to the original rectangle — only the shape has shifted.
Area of a triangle
The area of a triangle is half the base times the perpendicular height. Drop a straight line from the apex down to the base — that's the height, never the slanted edge.
Example: A triangle with a 10 cm base and a 6 cm perpendicular height has an area of ½ × 10 × 6 = 30 cm².
Any triangle fits inside a rectangle that's twice its area, which is where the ½ comes from. Measuring the slanted side instead of the perpendicular height is the single most common mistake on scholarship area papers.
Area of a circle
The area of a circle is pi times the radius squared. Radius is the distance from the centre to the edge — half the diameter. If a question gives the diameter, you have to halve it before squaring.
Example: A circle with a 5 cm radius has an area of π × 5² = π × 25 ≈ 78.54 cm².
π is approximately 3.14159. For scholarship working, keep π as a symbol throughout the calculation and only substitute the decimal at the very last step — examiners want to see it carried through.
Area of a trapezium
A trapezium has two parallel sides of different lengths. Its area is half the sum of those parallel sides, multiplied by the perpendicular height.
Example: A trapezium with parallel sides of 8 cm and 5 cm and a height of 4 cm has an area of ½ × (8 + 5) × 4 = 26 cm².
Think of it as a rectangle whose width is the average of the two parallel sides. Trapeziums show up regularly on scholarship papers from Year 6 onward.
Area of a rhombus
A rhombus has four equal sides, but its area depends on its two diagonals. The diagonals always meet at right angles, which is why this formula works.
Example: A rhombus with diagonals of 8 cm and 6 cm has an area of ½ × 8 × 6 = 24 cm².
You could also use base × height like a parallelogram, but diagonals are usually the easier measurement to read off a diagram.
Area of a regular polygon
A regular polygon (all sides and angles equal) divides into n identical triangles from the centre. The general formula is A = ¼ × n × s² × cot(π/n), where n is the number of sides and s is the side length.
Example: A regular hexagon (n = 6) with 4 cm sides has an area of about 41.57 cm².
You'll rarely use this formula directly in primary school, but scholarship and senior maths exams do test it. Keep the formula visible in the interactive tool above for reference.
Area is part of the measurement and geometry strand across Years 4–7 in the Australian Curriculum. Comprehensive practice is the fastest way to build the recognition skill that scholarship exams reward.
Worked example — find the area of a trapezium
Here's the five-step process in action. A trapezium has parallel sides of 8 cm and 5 cm, with a perpendicular height of 4 cm. Find its area.
Step 1 — identify the shape
Two parallel sides of different lengths — that's a trapezium.
Step 2 — pick the formula
Trapezium area is A = ½ × (a + b) × h, where a and b are the two parallel sides and h is the perpendicular height.
Step 3 — find the dimensions
From the question: a = 8 cm (longer parallel side), b = 5 cm (shorter parallel side), h = 4 cm (perpendicular height).
Step 4 — substitute and calculate
A = ½ × (8 + 5) × 4 = ½ × 13 × 4 = 26.
Step 5 — label the units
Lengths were in centimetres, so the area is in square centimetres. The trapezium has an area of 26 cm².
Three mistakes kids make (and how to fix them)
Forgetting to square the unit
Area is always measured in square units — cm², m², mm² — never plain cm or m. The exponent is part of the answer. Scholarship markers deduct marks for missing it, even when the number is right.
Using diameter when the formula needs radius
Circle area is π × r², not π × d². If the question gives the diameter, halve it first to get the radius, then square that. The same mistake — squaring d when r was needed — quadruples the answer.
Measuring the slanted side instead of the perpendicular height
For triangles, parallelograms, and trapeziums, the height in the formula is the perpendicular distance — the straight-up-and-down line, not the slanted edge. Look for the little right-angle square in the diagram; that marks the true height.
Words to know
These are the words your child will hear and need to use confidently when answering area questions.
- Area
- The amount of surface inside a 2D shape, measured in square units like cm² or m².
- Perimeter
- The total length of the outside of a shape — the distance you'd walk if you traced its edges.
- Base
- The bottom side of a triangle, parallelogram, or trapezium when measuring its area.
- Perpendicular height
- The straight-up-and-down distance from the base to the opposite vertex or side — never the slanted edge.
- Radius
- The distance from the centre of a circle to its edge. Half the diameter.
- Diameter
- The distance across a circle through its centre. Twice the radius.
- Circumference
- The distance around a circle — its perimeter.
- π (pi)
- The ratio of a circle's circumference to its diameter, approximately 3.14159. Use π exactly when you can; substitute 3.14 only at the final step.
- Square unit
- The unit area uses — a square measuring 1 unit on each side. Written cm² (square centimetres, also called 'cm squared'), m² (square metres or 'm squared'), or mm² (square millimetres). 1 cm² is a square 1 cm on each side.
- Parallelogram
- A four-sided shape with two pairs of parallel sides. Rectangles and rhombuses are special parallelograms.
- Trapezium
- A four-sided shape with exactly one pair of parallel sides.
- Apothem
- The distance from the centre of a regular polygon to the midpoint of one of its sides — the polygon equivalent of a radius.
Frequently asked questions
How do you find the area of a rectangle?
Multiply the length by the width: A = l × w. A rectangle that's 6 cm long and 4 cm wide has an area of 24 cm². Both measurements must be in the same unit, and the answer is always in square units (cm², m², mm²).
How do you find the area of a square?
A square is a rectangle with every side the same length, so you square the side: A = s². A square with 7 cm sides has an area of 49 cm². The 'squared' name comes from this exact relationship — multiplying a number by itself gives a square's area.
How do you find the area of a triangle?
Multiply the base by the perpendicular height and divide by two: A = ½ × b × h. A triangle with a 10 cm base and 6 cm perpendicular height has an area of 30 cm². The 'height' must be perpendicular (straight up and down) from the base — never the slanted edge of the triangle.
What's the formula for the area of a circle?
Pi times the radius squared: A = π × r². A circle with a 5 cm radius has an area of π × 25 ≈ 78.54 cm². Radius is the distance from the centre to the edge — half the diameter. If a question gives the diameter, halve it first before squaring.
How do you find the area of a trapezium?
Add the two parallel sides, multiply by the perpendicular height, then halve: A = ½ × (a + b) × h. A trapezium with parallel sides of 8 cm and 5 cm and a height of 4 cm has an area of ½ × 13 × 4 = 26 cm². The formula effectively averages the two parallel sides and multiplies by the height.
What's the difference between area and perimeter?
Perimeter is the distance around the outside of a shape — measured in plain units like cm. Area is the surface inside the shape — measured in square units like cm². Two shapes can share a perimeter but have very different areas (a long thin rectangle vs. a square with the same edge total).
Why does the height in the formula have to be perpendicular?
Because the area formulas for triangles, parallelograms and trapeziums all measure how 'tall' the shape is, straight up and down from the base — not the length of the slanted side. Diagrams usually mark the perpendicular height with a tiny right-angle square so students can spot it at a glance. Using the slanted edge instead of the perpendicular height is the single most common mistake in scholarship area questions.
Which year level is this tool for?
Years 4–7 specifically — Year 4 is when rectangles and triangles first appear, Years 5 and 6 cover circles, parallelograms, and trapeziums, and Year 7 is the main scholarship-exam year. Scholarship-leaning families use it for ACER, Edutest, AAS and ACER HAST preparation.