Circumference of a Circle: How to Use Pi, Radius, and Diameter

If a circle has a diameter of 10 centimetres, multiply 10 by pi. Using 3.14 as an approximation gives a circumference of about 31.4 centimetres. If the same circle is described by its radius, the radius is 5 centimetres. Using 2 times pi times 5 gives the same result, as it should.

The two formulas for circumference of a circle are really the same relationship written in different ways. Diameter equals 2 times radius. Substituting 2 times radius for the diameter turns pi times diameter into 2 times pi times radius. You do not need to memorise two unrelated rules if you understand that connection.

Pi appears because the ratio of a circle's circumference to its diameter is constant for every circle. Measure the distance around a circle and divide by its diameter, and the result approaches pi. This relationship is why circles of different sizes all use the same number in the circumference formula.

A frequent error is mixing up circumference of a circle with area of a circle. Circumference is a length, so the answer uses ordinary length units such as centimetres or metres. Area measures the space inside the circle, so it uses square units. If your circumference answer ends in square centimetres, that is a sign to check the formula and units.

Another mistake is using a radius value where the formula expects diameter, or using diameter where it expects radius. Circle diagrams sometimes label a line from the centre to the edge, which is a radius, and sometimes a line all the way across through the centre, which is a diameter. Identify which measurement you have before calculating.

You can also estimate before using a calculator. Since pi is a little more than 3, the circumference should be a little more than three times the diameter. A circle with diameter 8 should therefore have a circumference a little above 24. If your calculator gives 6 or 200, the estimate tells you immediately that something went wrong. This quick check makes circumference problems much safer. Reverse problems use the same relationships. If you know the circumference and need the diameter, divide the circumference by pi. If you need the radius, divide by 2 times pi. Keeping the units visible helps here too. A circumference measured in metres produces a diameter and radius in metres. This is a good reminder that the formula describes one geometric relationship that can be rearranged, not a one-way calculation you can only use from radius to circumference. If a problem gives a semicircle or another partial arc, do not automatically use the whole circumference. Work out what fraction of the circle is present, then include any straight edges if the question asks for total perimeter.

Here is a puzzle that surprises almost everyone. Imagine a rope stretched tightly around the Equator, roughly 40000 kilometres long. Now add just one metre to its length and lift it evenly so it floats above the ground all the way round. How high is the gap? Most people guess something microscopic, but the gap is about 16 centimetres, enough for a cat to squeeze under. The same would be true for a rope around a football: add one metre and the gap is still about 16 centimetres.

The formula explains why. Circumference equals 2 times pi times radius, so any increase in circumference equals 2 times pi times the increase in radius. Rearranging, the extra radius is the extra length divided by 2 times pi. One metre divided by about 6.28 is roughly 0.16 metres, whatever size the circle started at. The same reasoning is used in everyday measuring. A bicycle wheel with a diameter of 70 centimetres travels about 2.2 metres every time it turns, which is how cycle computers and old-style measuring wheels convert a count of rotations into a distance.

More than two thousand years ago, the Greek scholar Eratosthenes, working in Alexandria, estimated the circumference of the entire Earth using shadows and the idea of a fraction of a circle. He had learned that at Syene, far to the south, the midday Sun on the summer solstice shone straight down, so upright objects cast no shadow. At noon on the same date in Alexandria, a vertical pointer did cast a shadow, showing that the Sun's rays arrived about 7.2 degrees away from the vertical.

Because the Sun is so far away, its rays reach both cities almost parallel, so that 7.2 degree difference must equal the angle between the two places measured at the centre of the Earth. Since 7.2 is one fiftieth of 360, the distance from Alexandria to Syene, reckoned at about 5,000 stadia, should be one fiftieth of the full circumference. Multiplying by 50 gave 250,000 stadia. Historians still debate exactly how long his stadion was, so his accuracy is uncertain, but on plausible values his answer lands within a modest margin of the true figure of about 40,000 kilometres.

How do we know that pi is about 3.14? Archimedes, working in the third century BC, trapped the circumference between two polygons. Draw a regular hexagon inside a circle of radius 1. Each of its six sides equals the radius, so its perimeter is 6, and because the circle bulges outside it, the circumference must be more than 6, so pi is more than 3. A square drawn around the outside of the same circle has sides of 2 and a perimeter of 8, so pi must be less than 4. Polygons with more sides squeeze those two limits closer together.

Archimedes doubled the number of sides again and again, from 6 to 12, 24, 48 and finally 96, working out square roots by hand at each stage. He showed that pi lies between 223/71 and 22/7, roughly 3.1408 and 3.1429. That upper limit explains a common misconception. Many people believe pi equals 22/7, but 22/7 is 3.142857 and so on, while pi is 3.14159 and so on. The fraction is a handy approximation, accurate to about one part in 2,500, not the true value. In fact no fraction of whole numbers can express pi exactly, which is why its decimal digits never settle into a repeating pattern.

The circumference of a circle is the distance around the circle. Use pi times diameter when you know the diameter, or 2 times pi times radius when you know the radius. Keep the answer in length units, remember that diameter is twice the radius, and use the rough rule of a little more than three diameters as a fast sense check.