Radius of a Circle: Definition, Formula, Examples

TL;DR

The radius of a circle is the distance from its centre to any point on the boundary, and it is exactly half the diameter. This article covers the definition, the three formulas for finding the radius, how it ties to the circle equation, six worked examples, and the common mistakes.

One Length That Builds the Entire Circle

Fix a pin in a sheet of paper, tie a string to it, stretch the string tight, and swing a pencil all the way round. The shape you get is a perfect circle, and the length of that string never changed once. That single fixed length is the radius, and every other number you can measure about the circle, its width across, its distance round, the space it covers, comes straight out of it.

Once you see the radius as the one length the whole circle is built from, the formulas stop being a list to memorise and start being three views of the same idea.

What Is the Radius of a Circle?

The radius of a circle is the distance from the centre of the circle to any point on its boundary. We write it as the lowercase letter r. Pick the centre, pick any point on the edge, and the straight line joining them is a radius.

A circle has an endless number of radii, because there are endless points on its boundary, and every one of those radii is the same length. That equal-distance idea is the whole definition of a circle: the set of all points that sit the same distance from one centre point. The radius is that distance.

The radius is closely tied to two other measurements you will meet on every circle: the diameter, the full distance across the circle through the centre, and the circumference, the distance all the way around. The diameter is a chord that happens to pass through the centre, and it is always exactly twice the radius.

The Radius Formula — Three Ways to Find It

There is no single radius formula, because the radius can be recovered from whatever you happen to know about the circle. Each formula below is just one of the circle's measurements solved back for r. Define the variables once: r is the radius, d the diameter, C the circumference, A the area, and π (pi) is the constant ≈ 3.14159 that links a circle's distance round to its width across.

From The Diameter

The diameter runs through the centre and is made of two radii laid end to end, so it is twice the radius. Reverse that and the radius is half the diameter:

[ r = \frac{d}{2}. ]

From The Circumference

The distance round a circle is ( C = 2\pi r ), because the circumference is always π times the diameter and the diameter is 2r. Solving that for the radius:

[ r = \frac{C}{2\pi}. ]

From The Area

The space inside a circle is ( A = \pi r^2 ). To pull the radius back out, divide by π and take the square root:

[ r = \sqrt{\frac{A}{\pi}}. ]

How Do You Find The Radius of a Circle?

The method is always the same: identify which measurement you are given, then use the matching formula above. If you know the diameter, halve it. If you know the circumference, divide by 2π. If you know the area, divide by π and take the square root.

Two less common cases turn up in exams and are worth naming. If a circle is drawn on a coordinate grid with centre (h,k) and you know a point (x,y) on its boundary, the radius is the straight-line distance between them, found with the distance formula:

[ r = \sqrt{(x - h)^2 + (y - k)^2}. ]

And if a problem gives a chord of length c together with the perpendicular distance from the centre, the radius drops out of the Pythagorean theorem, because the radius, half the chord, and that perpendicular form a right triangle. We work both of these below.

The Radius in the Equation of a Circle

The radius is the heart of the equation of a circle. A circle centred at (h,k) with radius r is written:

[ (x - h)^2 + (y - k)^2 = r^2. ]

Every point (x,y) that satisfies this equation sits exactly r away from the centre, which is precisely the definition of the circle. So if you are handed an equation in this form, the radius is the square root of the number on the right.

Examples of the Radius of a Circle

Example 1: A circle has a diameter of 10 cm. Find its radius.

The radius is half the diameter, so:
[ r = \frac{10}{2} = 5 \text{ cm}. ]

Example 2: A circle has a circumference of 44 cm. Find its radius. Use π=( \frac{22}{7} ).

[ r = \frac{44}{2 \times \frac{22}{7}} = 7 \text{ cm}. ]

Example 3: A circle has an area of 78.5 cm². Find its radius. Use π=3.14.

[ r = \sqrt{\frac{78.5}{3.14}} = 5 \text{ cm}. ]

Example 4: A circle has an area of 616 cm². Find its radius. Use π=( \frac{22}{7} ).

[ r = \sqrt{616 \times \frac{7}{22}} = 14 \text{ cm}. ]

Example 5: A circle on a coordinate grid has its centre at (2,3) and passes through the point (5,7). Find its radius.

[ r = \sqrt{(5 - 2)^2 + (7 - 3)^2} = 5 \text{ units}. ]

Example 6: A chord of a circle is 16 cm long, and the perpendicular distance from the centre to that chord is 6 cm. Find the radius.

[ r^2 = 8^2 + 6^2 \rightarrow r = 10 \text{ cm}. ]

Where the Radius Shows Up

The radius matters because it is the one input that controls everything else about a circle, and that makes it the design parameter engineers reach for first.

Common Errors When Working With the Radius

Mistake 1: Halving the circumference instead of the diameter

Mistake 2: Forgetting the square root when working from the area

Mistake 3: Squaring the radius unit instead of keeping it linear

Key Takeaways

The radius of a circle is the distance from the centre to any point on the boundary, and it is exactly half the diameter.
The radius can be found three ways: ( r = \frac{d}{2} ) from the diameter, ( r = \frac{C}{2\pi} ) from the circumference, and ( r = \sqrt{\frac{A}{\pi}} ) from the area.
On a coordinate grid, the radius is the distance formula from the centre to a boundary point, and it is the square root of the right-hand side of the circle equation.
The most common mistake is halving the circumference; only the diameter is halved.
A circle has infinitely many radii, all equal in length, and the radius is always a positive length measured in linear units.

Practice These Problems to Solidify Your Understanding

A circle has a diameter of 18 m. Find its radius.
A circle has a circumference of 88 cm. Find its radius (use π=( \frac{22}{7} )).
A circle has an area of 154 cm². Find its radius (use π=( \frac{22}{7} )).

Answer to Question 1: r = 9 m. Answer to Question 2: r = 14 cm. Answer to Question 3: r = 7 cm.