Circle Calculator

About the Circle Calculator

Circles appear in almost every branch of mathematics and in countless practical problems: the length of fencing around a round garden, the amount of pizza in a slice, the distance a wheel travels in one turn or the cross-section of a pipe. All of these come down to a handful of formulas linking the radius to the diameter, circumference and area.

This calculator accepts whichever measurement you have. When you give the radius or diameter as an exact number, it keeps π symbolic, so a radius of 5 gives a circumference of exactly 10π as well as its decimal value of about 31.416. When you start from the circumference or area, it works backwards to the radius first and then finds everything else.

What are the parts of a circle?

A circle is the set of points at a fixed distance, the radius $r$ , from a centre. The diameter $d = 2r$ is the longest chord, passing through the centre. The circumference $C$ is the distance around the circle, and the area $A$ is the space it encloses.

The number $\pi \approx 3.14159$ is defined as the ratio of any circle's circumference to its diameter, so $C = \pi d = 2\pi r$ . Cutting a circle into thin wedges and rearranging them into a near-rectangle of width $\pi r$ and height $r$ shows why the area is $A = \pi r^2$ .

How to use the Circle Calculator

Choose which measurement you know: radius, diameter, circumference or area.
Enter its value; fractions such as 7/2 and decimals are accepted.
Optionally enter a central angle in degrees to get an arc length and sector area.
Press Calculate circle.
Read the exact answers in terms of π and their decimal values.

Formulas

Diameter

d = 2r

Circumference

C = 2\pi r = \pi d

Area

A = \pi r^2

Radius from area

r = \sqrt{A / \pi}

Arc length and sector

s = \tfrac{\theta}{360^\circ} \cdot 2\pi r,\quad A_s = \tfrac{\theta}{360^\circ} \cdot \pi r^2

Worked example: Radius 5

The worked example has a radius of 5. The diameter is $2 \times 5 = 10$ , the circumference is $2\pi \times 5 = 10\pi \approx 31.416$ and the area is $\pi \times 5^2 = 25\pi \approx 78.540$ .

Adding a central angle works the same way. For a circle of radius 6 and an angle of 60°, which is one sixth of the full turn, the arc length is $\frac16 \times 12\pi = 2\pi \approx 6.283$ and the sector area is $\frac16 \times 36\pi = 6\pi \approx 18.850$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

r = 5

How we read your problem

Find the radius, diameter, circumference and area of a circle from any one of them.

r = 5

Step-by-step solution

Diameter
$d = 2r = 10$
Circumference
$C = 2\pi r = 2\pi \cdot 5 = 10\pi \approx 31.41592654$
Area
$A = \pi r^2 = \pi \cdot 5^2 = 25\pi \approx 78.53981634$

Formulas used

Diameter
$d = 2r$
Circumference
$C = 2\pi r = \pi d$
Area
$A = \pi r^2$
Arc length
$s = \frac{\theta}{360^\circ} 2\pi r$

Every measurement of a circle follows from its radius. The circumference grows in proportion to the radius and the area in proportion to its square.

Verification

C/d = 3.141592654 = π.
A = C·r/2 = 78.53981634.

Detailed explanation

Working backwards from circumference or area

If you know the circumference, divide by $2\pi$ to get the radius. A circular running track with a circumference of 400 m has a radius of $400 / (2\pi) \approx 63.66$ m. From there, the enclosed area is about 12,732 m².

If you know the area, divide by π and take the square root. A circular rug with an area of 50 square feet has a radius of $\sqrt{50/\pi} \approx 3.989$ ft. Because these results involve dividing by π, they are irrational in general, so the calculator gives them as decimals.

Exact answers in terms of π

Many teachers and exams ask for answers "in terms of π". That means leaving π as a symbol rather than replacing it with 3.14. A circle with diameter 7 has radius $\frac72$ , circumference $7\pi$ and area $\frac{49}{4}\pi$ . These exact forms avoid rounding error and make later algebra cleaner.

When an approximate answer is required, use the π key on your calculator rather than 3.14 or 22/7. Those approximations are good to about two or three significant figures, which can visibly change answers for large circles.

How radius changes area

Circumference is proportional to the radius, but area is proportional to its square. Doubling the radius doubles the circumference and quadruples the area. This is why a 16-inch pizza has almost twice the area of a 12-inch pizza, since $(16/12)^2 \approx 1.78$ .

The ratio $C/d$ is the same for every circle, which is the defining property of π. The calculator reports this ratio as a check, along with the identity $A = \frac12 C r$ , which connects the two main formulas.

Common mistakes

Using the diameter in $A = \pi r^2$ without halving it first.
Confusing circumference with area and giving area in plain units instead of square units.
Rounding π to 3.14 too early in a multi-step problem.
Forgetting to take the square root when finding the radius from an area.
Using a central angle in radians in a degree formula.

Tips

Circumference is a length (cm, m); area is in squared units (cm², m²).
If a problem gives a diameter, write down the radius first.
Leave answers in terms of π unless a decimal is asked for.
A semicircle's perimeter is half the circumference plus the diameter, not just half the circumference.
Use the area calculator for other shapes and the volume calculator for cylinders, cones and spheres.

Frequently asked questions

How do I find the circumference of a circle?

Multiply the diameter by π, or twice the radius by π: C = πd = 2πr. A radius of 5 gives C = 10π ≈ 31.42.

How do I find the radius from the circumference?

Divide the circumference by 2π: r = C / (2π). A circumference of 100 gives r ≈ 15.92.

How do I find the radius from the area?

Divide the area by π and take the square root: r = √(A/π). An area of 50 gives r ≈ 3.989.

What does "in terms of π" mean?

It means leaving π as a symbol in the answer, such as 25π instead of 78.54. The result is exact rather than rounded.

How do I find the arc length of a sector?

Multiply the circumference by the fraction of the circle the angle covers: s = (θ/360°) × 2πr for θ in degrees, or s = rθ for θ in radians.

Why does doubling the radius quadruple the area?

Area depends on r², so replacing r by 2r multiplies the area by 2² = 4. The circumference, which depends on r, only doubles.

What is the difference between a chord and a diameter?

A chord is any segment joining two points on the circle. A diameter is the special chord that passes through the centre, and it is the longest possible chord.

Measure your circle

Enter any one measurement above to get the radius, diameter, circumference and area, exact and in decimals.

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