What is the area of a circle with radius 13?

The area is 169π, which is 530.929158 square units. The formula is A = πr², so squaring the radius gives 13 × 13 = 169, and multiplying by π gives 530.929158. Because the radius squares, the area of this circle is 169 times that of the unit circle rather than 13 times.

What is the circumference of a circle with radius 13?

The circumference is 26π, or 81.681409 units. It comes from C = 2πr, so 2 × 13 = 26, then × π. The same circle has a diameter of 26, and since C = πd you can get there either way — 26π is the same as 26π. Dividing the circumference by the diameter returns π itself, 3.14159265, which is true for every circle and is where the constant comes from.

How much bigger is radius 13 than radius 12?

The radius goes up by 8.33%, from 12 to 13, but the area goes up by 17.36% — from 452.389342 to 530.929158. That is the square law: the area ratio is (13/12)² = 1.173611. The circumference, being linear, rises by the same 8.33% as the radius.

What fits inside a circle of radius 13?

A square inscribed in it has a diagonal equal to the diameter, 26, so its side is 13√2 = 18.384776 and its area is 2r² = 338 — about 63.7% of the circle, a proportion that is the same for every circle. A regular hexagon inscribed in it has a side of exactly 13, since the hexagon side always equals the radius, giving a perimeter of 78 against a circumference of 81.681409.

What is the difference between a chord, an arc and a segment?

A chord is a straight line joining two points on the circle; the diameter is the longest one there is. An arc is the curved part of the circumference between those same two points. A segment is the region between a chord and its arc, whereas a sector is the region between two radii and their arc. The distinction matters because the formulas are quite different.

Why is 360 degrees a full turn?

Convention inherited from Babylonian astronomy, which counted in sixties and used a 360-day year. It survives because 360 divides evenly by so much — 2, 3, 4, 5, 6, 8, 9, 10, 12 and more — which makes common fractions of a circle come out as whole numbers of degrees. Mathematics itself prefers radians, where a full turn is 2π, because that is the unit in which the calculus of circles behaves.

What are the formulas for a circle?

Three, and they all hang off the radius. The diameter is d = 2r. The circumference is C = 2πr, which is the same as πd. The area is A = πr². If you have any one of the four quantities you can recover the other three, which is what the mode buttons on this calculator do — enter a circumference or an area and it works backwards to the radius first.

Why does the area grow so much faster than the circumference?

Because area depends on the square of the radius and circumference only on the radius itself. Double the radius and the circumference doubles while the area quadruples; triple it and the area goes up ninefold. It is why a pizza twice the width is four times the food, and why the perimeter of a field grows far more slowly than the land inside it as the field gets bigger.

The ratio of any circle’s circumference to its diameter — the same number for every circle, which is the surprising part and the reason it earned a name. It is irrational, so it has no exact decimal or fraction, and transcendental, which is what makes squaring the circle with compass and straightedge impossible. To eight places it is 3.14159265.

Circle with Radius 13

A circle of radius 13 has an area of 530.929158 and a circumference of 81.681409, with a diameter of 26. Both of those are exact multiples of π before they are turned into decimals — the area is 169π and the circumference is 26π — which is worth seeing, because it is where the irrationality lives. N...

CIRCLE CALCULATOR

SOLVE FROM

Radius (r)

Must be greater than 0

Central Angle θ (optional) — for arc length & sector area

RADIUS (r)

units

DIAMETER (d)

units

CIRCUMFERENCE (C)

81.681409

units

AREA (A)

530.929158

units²

STEP-BY-STEP

Given: radius r = 13

Diameter: d = 2r = 2 × 13 = 26

Circumference: C = 2πr = 2 × π × 13 = 81.681409

Area: A = πr² = π × 13² = 530.929158

QUICK EXAMPLES

LIVE DIAGRAM

EXACT EXPRESSIONS

Circumference13π × 2 = 81.681409

Areaπ × 13² = 530.929158

C/d ratioπ = 3.14159265...

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HOW TO USE

1
The answer is already here: a circle of radius 13 has an area of 530.929158 and a circumference of 81.681409. This page opens with the radius set to 13, so nothing needs typing.
2
Read the four result tiles — radius 13, diameter 26, circumference 81.681409 and area 530.929158. The STEPS panel underneath shows each one being derived rather than just asserted.
3
Add an angle to get a wedge. A 90° sector of this circle has an arc of 20.420352 and an area of 132.73229; a 60° sector has an arc of 13.613568 and an area of 88.488193.
4
Switch the input mode to run the same circle backwards — enter the area 530.929158 or the circumference 81.681409 and it recovers the radius 13, which is a useful check that everything is consistent.

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Last updated: July 28, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.