Here are the main circle formulas: circumference, area, radius, diameter and the relationships between these values.
Circle formulas at a glance
- Circumference of a circle: \(C=2 \times \pi \times r=\pi \times d\)
- Area of a circle: \(S=\pi \times r^{2}=\frac{\pi \times d^{2}}{4}\)
- Diameter: \(d=2r\)
- Radius from diameter: \(r=\frac{d}{2}\)
- Radius from circumference: \(r=\frac{C}{2\pi}\)
- Diameter from circumference: \(d=\frac{C}{\pi}\)
A circle is the part of a plane bounded by a circumference. The circumference is the boundary of the circle: all points at the same distance from the center.
The radius is the distance from the center to the circumference. The diameter is a line segment through the center that connects two points on the circumference. The diameter is twice the radius.
Circumference of a circle
The circumference of a circle is the length of its boundary.
$$C=2 \times \pi \times r= \pi \times d$$
where,
C - circumference;
π - pi, approximately 3.14159;
r - radius of the circle;
d - diameter of the circle.
Example: if the radius is 5 cm, then \(C=2 \times \pi \times 5=10\pi\approx31.42\) cm.
Area of a circle
The area of a circle is the size of the plane region bounded by the circumference.
$$S=\pi \times r^{2}=\frac{\pi \times d^{2}}{4}$$
where,
S - area of the circle;
r - radius of the circle;
d - diameter of the circle.
Example: if the radius is 5 cm, then \(S=\pi \times 5^{2}=25\pi\approx78.54\) cm2.
Using the diameter: if \(d=10\) cm, then \(S=\frac{\pi \times 10^{2}}{4}=25\pi\approx78.54\) cm2.
Radius and diameter
The diameter is two radii:
$$d=2r$$
The radius is half the diameter:
$$r=\frac{d}{2}$$
Example: if the diameter is 12 cm, then the radius is \(r=\frac{12}{2}=6\) cm.
Finding radius and diameter from circumference
If the circumference is known, both the radius and diameter can be found from it.
$$r=\frac{C}{2\pi}$$
$$d=\frac{C}{\pi}$$
Example: if the circumference is 31.42 cm, then \(r=\frac{31.42}{2\pi}\approx5\) cm and \(d=\frac{31.42}{\pi}\approx10\) cm.
Semicircle area and perimeter
The area of a semicircle is half the area of a circle with the same radius:
$$S_{semicircle}=\frac{\pi r^{2}}{2}$$
The perimeter of a semicircle including the diameter:
$$P_{semicircle}=\pi r+2r$$
Value of pi
π is the ratio of a circle's circumference to its diameter. Calculations often use \(\pi\approx3.14\), more precisely \(\pi\approx3.14159265359\).
Frequently asked questions
How do you calculate the circumference of a circle?
If the radius is known, use \(C=2\pi r\). If the diameter is known, use \(C=\pi d\).
How do you calculate the area of a circle?
If the radius is known, use \(S=\pi r^{2}\). If the diameter is known, use \(S=\frac{\pi d^{2}}{4}\).
What is the diameter of a circle?
The diameter is a line segment through the center that connects two points on the circumference. The diameter is two radii: \(d=2r\).
Does a circle have volume?
A circle is a plane figure, so it has no volume. Volume is calculated for three-dimensional solids, such as a sphere or cylinder.
What is the difference between a circle and a circumference?
A circle is the plane region inside the boundary. The circumference is only the boundary line.
