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Area of a Circle

In one line: The area of a circle is the amount of flat space inside it, and you find it with A = πr², which means pi times the radius times the radius.

Why it works

Circles are everywhere: pizzas, garden ponds, round tables, sprinkler patterns and wheels. Knowing the area tells you how much paint, grass seed, fabric or pizza you are dealing with. A few words first. The radius (r) is the distance from the centre of the circle to its edge. The diameter is the distance straight across through the centre, so it is twice the radius. Pi (π) is a special number, about 3.14159, that is the same for every circle. The area formula also comes up again later in maths, in the volume of cylinders and cones and the surface area of spheres.

Think of it like: Imagine cutting a round pizza into lots of thin slices, like a sector fan. Now lay the slices in a row, pointing up, then down, then up, fitting them together. The shape you get looks almost like a rectangle. Its height is the radius, r. Its length is half the crust, because half the slices point up and half point down. The whole crust (the circumference) is 2πr, so half of it is πr. The area of a rectangle is length × height, so the area is πr × r = πr². (Where this breaks: The rearranged slices never make a perfect rectangle. The top and bottom edges are still a little bumpy because the crust is curved. The more slices you cut, the closer it gets to a true rectangle. The formula is exactly right, but the pizza picture only shows why it makes sense; it is not a proof by itself.)

Worked example

A round rug has a diameter of 10 cm on a scale drawing. Find its area, correct to 2 decimal places.

  1. Find the radius: the radius is half the diameter, so r = 10 ÷ 2 = 5 cm.
  2. Write the formula: A = πr².
  3. Square the radius first: r² = 5 × 5 = 25.
  4. Multiply by pi: A = π × 25 = 25π.
  5. Use the π button on a calculator: 25π ≈ 78.54.
  6. Add square units, because area measures flat space: cm².

Answer: The area is 25π cm², which is about 78.54 cm².

Common mistakes

  • Putting the diameter into the formula instead of the radius gives an area four times too big, so always halve the diameter first.
  • Using 2πr finds the circumference (the distance around the edge), not the area, so check that your formula has r squared.
  • Working out (π × r)² squares pi as well, so square only the radius and then multiply by π.
  • Writing the answer in cm instead of cm² hides that area is a measure of flat space, so always use square units.

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A circle has a radius of 3 cm. What is its area, to 2 decimal places?

Questions parents ask

What year do kids learn the area of a circle?

In Australia it is usually taught in Year 8, and in the UK it often appears in Year 7 or Year 8. In the US it is part of Grade 7 maths. Students then use it again in later years for cylinders, cones and spheres.

How can I help my child with the area of a circle at home?

Measure round things together, such as plates, lids or a pizza, and work out the area from the radius. Ask your child to explain each step out loud: halve the diameter, square the radius, multiply by π, then add square units. Cutting a paper circle into slices and rearranging them into a near-rectangle is a great way to show why the formula works.

Should my child use 3.14 or the π button?

Both are fine if the question says which to use. The π button is more accurate, so answers can differ slightly from those using 3.14 or 22/7. Sometimes the best answer is left in terms of π, such as 25π cm², which is exact.

Why is the answer in square units?

Area measures how many small squares would cover a flat shape, such as squares 1 cm by 1 cm. Because the radius is multiplied by itself, the units are multiplied too: cm × cm = cm². A length like the circumference stays in plain cm.