Diameter of a circle

The diameter of a circle is a straight line that goes from one side of the circle to the other, passing through the centre (center in the US). Its length is twice the radius, d = 2r. It is the longest straight line you can fit inside a circle, and you can also find it from the circumference, d = C ÷ π.

What is the diameter?

Draw any straight line across a circle through its centre and you have drawn a diameter. A circle has infinitely many diameters, one in every direction, and they are all the same length. The word means both the line and its length: "draw a diameter" and "the diameter is 10 cm" are both correct.

The same idea works for any round object. The diameter of a ball, a pipe, a coin or a planet is the distance straight across it through the middle. On engineering drawings, it is often marked with the symbol ⌀, so ⌀20 means a diameter of 20 mm.

Diameter, radius, circumference and chord

In England, AQA's GCSE Maths specification lists the centre, radius, chord, diameter and circumference among the circle parts students must know and use.

Diameter formulas

Pick the formula that starts from what you know.

π (pi) is about 3.14159. It is the number you get when you divide any circle's circumference by its diameter, which is exactly why C = πd. The US Common Core expects grade 7 students to know the circumference and area formulas and use them to solve problems, and AQA expects GCSE students to know C = 2πr = πd and A = πr².

Circumference against diameter

Here x is the diameter and y is the circumference, in the same units. The points lie on a line that starts at (0, 0) and rises with a slope (gradient) of π: double the diameter and the circumference doubles too. The area does not work like that. It goes up with the square of the diameter, so a circle twice as wide has four times the area.

How to find the diameter: worked examples

Example 1: from the radius

A circle has a radius of 7 cm. Its diameter is 2 × 7 = 14 cm.

Example 2: from the circumference

A circular table has a circumference of 50 cm. Find its diameter.

The diameter is 15.9 cm to 1 decimal place. Use the π button on your calculator rather than 3.14, then round at the end.

Example 3: from the area

A circle has an area of 113 cm². Find its diameter. First find the radius, then double it:

The diameter is 12.0 cm to 3 significant figures. A common slip is to divide the area by π and stop, which gives r², not r.

Example 4: on a coordinate grid

The ends of a diameter are at (−2, 1) and (6, 7). Find the length of the diameter, the centre and the radius.

• Length: the distance formula gives √((6 − (−2))² + (7 − 1)²) = √(64 + 36) = √100 = 10.
• Centre: the midpoint of the diameter, ((−2 + 6) ÷ 2, (1 + 7) ÷ 2) = (2, 4).
• Radius: half the diameter, 5.

So the circle has diameter 10, centre (2, 4) and radius 5, and its equation is (x − 2)² + (y − 4)² = 25. The midpoint and distance calculator does the first two steps for any pair of points.

Example 5: which pizza is bigger?

Pizza sizes are diameters. Is one 12 inch pizza more or less food than two 8 inch pizzas?

• One 12 inch pizza: radius 6, area π × 6² = 113.1 square inches.
• Two 8 inch pizzas: radius 4 each, area 2 × π × 4² = 100.5 square inches.

The single 12 inch pizza has about 12.5% more pizza. The diameter only grows by half, from 8 to 12, but the area grows with its square.

The diameter and the angle in a semicircle

Join both ends of a diameter to any other point on the circle and the angle you make at that point is always 90°. This is one of the circle theorems in AQA's GCSE Maths specification: the angle at the circumference in a semicircle is 90°. It gives a quick test, too: if a triangle drawn inside a circle has one side as a diameter, it is a right-angled triangle (right triangle in the US).

Diameter of a sphere

A sphere's diameter is the distance straight through its centre, and it is still twice the radius. Written with the diameter, the volume of a sphere is V = ⅙πd³. A ball with a diameter of 10 cm has a volume of ⅙ × π × 1,000 = 524 cm³ to 3 significant figures.

Common mistakes

• Mixing up radius and diameter. Check which one the question gives you. A = πr² needs the radius, so halve a diameter first.
• Drawing a diameter that misses the centre. A line across the circle that doesn't pass through the centre is a chord, and it is shorter.
• Forgetting the square root when going from area to diameter.
• Rounding π too early. Use the π button and round only the final answer.
• Leaving out units. A diameter is a length, so it is in cm, m or inches, never cm².

To practise circles among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Foundation.

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