Why is the area of a circle πr²? Slice it and see

How to use this simulation

1. Start in “Area proof” mode. The circle is cut into 12 equal sectors, coloured in two alternating shades. Press Play and watch the pieces slide out and lock into each other like the teeth of two combs.
2. Drag the “Number of sectors” slider. With only 4 slices the shape is a bumpy parallelogram. With 64 it is almost a perfect rectangle. The readings show the rectangle’s width, its height, its area and the error against the exact πr².
3. Use the “Rearranging” slider to scrub the animation by hand, frame by frame, and stop at any moment to compare the circle with the strip.
4. Change the radius. Every length on the page scales, and the circumference, half circumference and area follow. Notice the error percentage does not depend on the radius at all, only on the number of slices.
5. Switch to “Circumference and pi”. A circle rolls along a line and leaves its circumference behind. The line is cut into diameters: three whole diameters and a little bit left over. That little bit is the 0.14159… of pi.
6. Switch to “Sector and arc” and drag the angle θ. The arc length and sector area are always the same fraction θ/360 of the whole circle, and the two bars at the bottom show that fraction filling up.

A pizza, a circle and a very old question

You order a 30 cm pizza cut into 8 slices. Your friend says the 40 cm one is “only a bit bigger”. Is it? The small pizza has an area of about 706.86 cm², so each slice is 88.36 cm². The large pizza has 1,256.64 cm², which is 1.78 times as much food for a pizza that is only a third wider. To see why, you need the area of a circle.

People have asked how to measure a round field for about four thousand years. Squares and rectangles are easy: multiply two sides. A circle has no straight sides at all, so for centuries the only honest answers were rough guesses. Around 250 BC the Greek mathematician Archimedes found a way to trap the answer between two numbers by cutting circles into more and more pieces. The simulation on this page uses the same idea of slicing, only with a rearrangement that makes the result easy to see.

The circle from zero: radius, diameter and circumference

A circle is every point at the same distance from one fixed point called the centre. Take a compass, stick the needle in the paper, turn the pencil all the way round, and the curve you draw is a circle. The size of the compass opening is the radius.

Everything about a circle is built from three lengths and one number. Once you are comfortable with them, the area formula stops being something to memorise.

• The radius r runs from the centre to any point on the circle.
• The diameter d runs straight across through the centre, so d = 2r.
• The circumference C is the length of the curve itself, the distance once around.
• The number π (pi) is the ratio C ÷ d, the same for every circle, big or small.

Pi: the number hiding in every circle

Take any round thing, a bottle cap, a plate, a bicycle wheel. Measure around it with a string, then measure across it. Divide the first by the second. You get about 3.1416 every time. That constant is pi, written π.

Switch the simulation to “Circumference and pi” mode and press Play. The circle rolls once around and lays its circumference flat. Compare that line with the circle’s diameter: three whole diameters fit, with a short piece left over. The leftover is roughly 0.14 of a diameter. Change the radius and the picture grows or shrinks, but the ratio in the readings does not move.

Because C = πd, and the diameter is two radii, the circumference is C = 2πr. That one fact is the first half of the area proof.

The slicing proof: why the area is πr²

Here is the proof the simulation draws. It uses nothing more than the circumference formula and the area of a rectangle.

A short history of pi

The Babylonians and Egyptians used values near 3.125 and 3.16 about 4,000 years ago. They measured; they did not prove.

Archimedes drew a polygon with 96 sides inside a circle and another outside it. The circle’s circumference had to lie between the two perimeters, which trapped pi between 223/71 (about 3.1408) and 22/7 (about 3.1429). That is where the fraction 22/7 comes from. It is a good approximation, not the real value.

In the 5th century the Chinese mathematician Zu Chongzhi found 355/113, which is correct to six decimal places. Around 1400, Madhava of Sangamagrama in Kerala used infinite series to get pi to many more digits. Today computers have calculated trillions of digits. Pi never ends and never repeats, which means it is not a fraction; 22/7 and 355/113 are only very good neighbours.

Try it in the simulation

Five small experiments turn the proof into something you can feel. Do them with the sliders, not just by reading.

• Set the number of sectors to 4, then to 12, then to 64. Watch the error drop. For r = 5 and n = 12, the rectangle’s area is 75 against the exact 78.54, an error of 4.51%.
• Set the radius to its smallest, then its largest. The shape changes size but never kind; the percentage error stays put. The error depends on the slicing, not the size.
• Double the radius from 5 to 10 cm in the readings. The circumference doubles but the area becomes four times larger (78.54 to 314.16 cm²). Area grows with the square of the radius.
• In “Circumference and pi” mode, read C ÷ d for radius 1, 6 and 12. It is the same number each time, which is exactly what makes pi a constant.
• In “Sector and arc” mode, set θ to 180. The arc is half the circumference and the sector is half the area: a semicircle. Set θ to 360 and the two bars fill completely.

Sector and arc: a slice of the formula

A sector is a slice of a circle with the point at the centre. If its central angle is θ degrees, it is the fraction θ/360 of the whole circle. So its area and its arc length are the same fraction of the circle’s area and circumference.

The fraction is the whole idea. A 90° sector is one quarter. A 60° sector is one sixth. A 180° sector is a half. Work out the fraction first and multiply.

Solved problems

Each answer below is computed from the same formulas the simulation uses, with π = 3.1416.

Common mistakes

These five slips cost marks every year. Check for them before you hand in a paper.

• Using the diameter in place of the radius in πr². Always halve the diameter first.
• Writing the answer in cm instead of cm². Area is flat space, so the unit is squared.
• Mixing up 2πr (circumference) and πr² (area). One has a 2 and no square; the other has a square and no 2.
• Subtracting the radii of a ring before squaring. The ring area is π(R² − r²), not π(R − r)².
• Forgetting the straight edges when asked for the perimeter of a semicircle or a sector.

Where circles show up in real life

Circle area decides how much pizza you get, how much paint covers a round table and how much water a circular pond holds. Farmers use it for centre-pivot irrigation fields, whose watered area is πr² with r the length of the sprinkler arm.

Engineers use it for pipes: the flow a pipe can carry depends on its cross-section area πr², so doubling the radius of a pipe quadruples what it carries. Cyclists use the circumference: one turn of the wheel moves the bicycle by 2πr, which is how a speedometer works.

Doctors compare areas of circular wounds to check healing, and astronomers estimate the area of the Sun’s disc. Every time something round matters, πr² is nearby.

Exam corner

Class 7 to 10 questions on circles are mostly one of five types. Learn to recognise the type before you start writing.

• Direct substitution: given r or d, find the area or the circumference.
• Reverse: given the area or the circumference, find r first.
• Composite shapes: a ring, a semicircle, a square with a circle cut out. Add or subtract areas.
• Sectors and arcs: write the fraction θ/360 first, then multiply.
• Cost and rate: fence uses circumference, tiling uses area; multiply by the price at the end.

Revision summary

If you remember nothing else from this page, remember these lines.

• The slicing proof gives width πr and height r, so area πr².
• Use π = 3.1416 unless the question says otherwise.
• Area is in squared units, circumference in plain units.

Frequently asked questions