How to Use the Calculator?
To use this Calculator, enter any one known value—radius, diameter, circumference, or area—and select the correct unit. The calculator will automatically find the remaining values for you.
You can also change the unit of any result using its dropdown, and the value will update automatically to match the selected unit.
What is the Radius of a Circle?
A circle is one of the most common shapes we see in everyday life, from the wheels of a bicycle to a coin, a pizza, or a clock. In mathematics, a circle is a closed shape where every point on the boundary is the exact same distance from a fixed center point. This equal distance from the center to the edge is known as the radius.
Suppose the center of a circle is point \(O\) and point \(A\) lies somewhere on the boundary. The straight line segment connecting point \(O\) to point \(A\) is the radius.
The radius is the most fundamental measurement of a circle because almost every other property and calculation depends on it.
Diameter of a Circle
The diameter is another essential part of a circle. It is the straight-line distance entirely across the circle, starting at one edge, passing directly through the center, and ending at the opposite edge.
Because the diameter spans the entire width of the shape, it is always exactly twice the length of the radius.
The mathematical relationship between the two is expressed by the formula:
\[d = 2r\]
Here, \(d\) represents the diameter and \(r\) represents the radius.
Example 1: Finding Diameter from Radius
Problem: Suppose a circle has a radius of \(7 \text{ cm}\). What is its diameter?
Solution: Using the diameter formula:
\[d = 2r\]
Substitute the given value of the radius:
\[d = 2 \times 7\]
\[d = 14 \text{ cm}\]
Therefore, the diameter of the circle is \(14 \text{ cm}\).
Example 2: Finding Radius from Diameter
Problem: The diameter of a circular garden is \(20 \text{ m}\). What is its radius?
Solution: Since the radius is exactly half of the diameter, you can rearrange the formula to solve for the radius:
\[r = \frac{d}{2}\]
Substituting the known diameter gives:
\[r = \frac{20}{2}\]
\[r = 10 \text{ m}\]
The radius of the circular garden is \(10 \text{ m}\).
Circumference of a Circle
The circumference is the total distance around the circle’s outer edge, which is very similar to the perimeter of a square or rectangle. Imagine taking a piece of string, wrapping it perfectly around the edge of a circular plate, and then straightening that string out against a ruler. That length is the exact circumference of the plate.
Because a circle is a continuous curve rather than a shape made of straight lines, we cannot calculate its boundary by simply adding side lengths together. Instead, we rely on a specific mathematical formula.
No matter how large or small a circle is, if you divide its circumference by its diameter, you will always get the exact same ratio: \(3.14159\)…. This special number is famously known as Pi, represented by the Greek letter \(\pi\).
Because of this constant relationship, calculating the circumference is straightforward. Using the radius, the formula for circumference is:
\[C = 2\pi r\]
In this equation, \(C\) is the circumference, \(r\) is the radius, and \(\pi\) is approximately \(3.14\). Since the circumference is directly proportional to the radius, bigger circles will naturally have larger circumferences.
Example 3: Finding Circumference Using Radius
Problem: A circular clock has a radius of \(6 \text{ cm}\). Find its circumference.
Solution: Using the formula for circumference:
\[C = 2\pi r\]
Substitute the known values:
\[C = 2 \times 3.14 \times 6\]
\[C = 37.68 \text{ cm}\]
The circumference of the clock is approximately \(37.68 \text{ cm}\).
Example 4: Finding Radius Using Circumference
Problem: A circular swimming pool has a measured circumference of \(31.4 \text{ m}\). What is its radius?
Solution: Start with the standard circumference formula:
\[C = 2\pi r\]
Rearrange it to solve for the radius:
\[r = \frac{C}{2\pi}\]
Substituting the pool’s circumference yields:
\[r = \frac{31.4}{2 \times 3.14}\]
\[r = \frac{31.4}{6.28}\]
\[r = 5 \text{ m}\]
The radius of the swimming pool is \(5 \text{ m}\).
Area of a Circle
Understanding the area of a circle starts with a simple question: how much flat space does the shape cover? If you imagine placing a circular plate on a table, the area is the exact amount of the table’s surface hidden beneath that plate.
Because of its unique, continuous curve, calculating the area requires a specific mathematical formula based on the radius.
The formula for area is:
\[A = \pi r^2\]
Here, \(A\) stands for area and \(r\) stands for the radius.
Because the radius is squared in this calculation, even a small increase in the length of the radius will cause the total area to increase greatly. Area is always measured in square units.
Example 5: Finding Area from Radius
Problem: A circle has a radius of \(5 \text{ cm}\). Find its total area.
Solution: Using the area formula:
\[A = \pi r^2\]
Substitute the value of the radius:
\[A = 3.14 \times 5^2\]
Squaring the radius first updates the equation to:
\[A = 3.14 \times 25\]
\[A = 78.5 \text{ cm}^2\]
The area of the circle is \(78.5 \text{ square centimeters}\).
Example 6: Finding Radius from Area
Problem: A large circular rug covers an area of \(113.04 \text{ m}^2\). What is its radius?
Solution: Begin with the area formula:
\[A = \pi r^2\]
Rearrange it to isolate the radius squared:
\[r^2 = \frac{A}{\pi}\]
Substituting the given area into the equation gives:
\[r^2 = \frac{113.04}{3.14}\]
\[r^2 = 36\]
To find the actual radius, take the square root of the result:
\[r = \sqrt{36}\]
\[r = 6 \text{ m}\]
The radius of the circular rug is \(6 \text{ m}\).
