Circle Formula Calculator

With this calculator you will be able to calculate the circumference and the area of a circle.

Circumference and area of circles are relatively simple to compute, provided that you give a valid radius, which in this case is a positive one.

You don't need to provide necessarily a number or a decimal, you can also provide fractions (ex: '2/3'), or any valid numeric expression, as long as it is non-negative.

What is the circle formula

There are several circle formulas depending on what you are trying to compute. For example, the simplest and more well known formulas are those for the area and the perimeter:

\[\text{Area} = \pi \cdot r^2 \] \[\text{Perimeter} = 2 \pi \cdot r \]

These formulas are fairly simple, as they just require for you to plug-in the value of \(r\) in them. Recall that \(\pi\) is just a constant that is approximately equal to \(\pi \approx 3.14159265359\)

Example: Area and Perimeter Calculation

Consider a circle with radius \(r = \frac{3}{4}\), then by just looking at the formulas above, and plugging into them the value of \(r = \frac{3}{4}\), we get that

\[\text{Area} = \pi \cdot r^2 = \pi \cdot \left( \displaystyle \frac{3}{4} \right)^2 = \pi \cdot \left( \displaystyle \frac{3^2}{4^2} \right)\] \[ = \pi \cdot \left( \displaystyle \frac{9}{16} \right) = \displaystyle \frac{9\pi }{16}\]

Then, you would report the area to be \( A = \displaystyle \frac{9\pi }{16}\). Now, the area may have units, depending on whether \(r\) was given with units. For example, if \(r = \frac{3}{4}\) feet, then the units for the area would be \(\text{feet}^2\), and you would report the area to be \( A = \displaystyle \frac{9\pi }{16} \text{ feet}^2\).

For the perimeter we get now:

\[\text{Perimeter} = 2 \pi \cdot r = 2 \pi \cdot r \left( \displaystyle \frac{3}{4} \right) = \displaystyle \frac{2 \cdot 3}{4} = \displaystyle \frac{3}{2} \]

where we can see in this case that we cancelled 2 both in the numerator and denominator. Then, you say that the perimeter is \(P = \displaystyle \frac{3}{2} \). Unlike the area case, the units of the perimeter are the same as the units of the radius.

So then, if for example the radius is measured in feet, then you would report that the perimeter is \(P = \displaystyle \frac{3}{2}\) feet.

Steps for Applying the Circle Formula

• Step 1: Identify whether you are looking at computing the area or perimeter, or maybe you are looking at finding both, so you would use both formulas
• Step 2: For the perimeter you use \(P = 2 \pi \cdot r \), and for the area you use \(A = \pi \cdot r^2 \)
• Step 3: For the given value of \(r\), you need to make sure that it is valid and positive. Then, you plug it into the formula
• Step 4: If the radius is given with units, the perimeter will have the same units as the radius, and the area will be "unit"², where "unit" is the unit for the radius

In the end, using the circle formulas is about making sure you have a valid radius \(r\), and plugging its value into the corresponding equation, making sure you report the correct units, if units are provided for \(r\).

Equation of Circle

When dealing with circle formulas, perhaps you are interested in actually computing the equation of a circle. There are many circumstances in which that could be the case.

For example, you may be given the center and the radius and you can directly get the equation of the corresponding circle. But also you could need to find a circle equation given a general quadratic equation. This is a lot more difficult and it involves completing the squares.

These operations can certainly be algebraically complex and error prone, so you be extremely careful and check your work frequently.

Do I really need to know the formula of a circle by heart?

The answer is: It depends. Often times, for more elementary courses, you will only need to use the formulas, and you may have a cheat sheet with the all the information you need to work with a circle.

Now, if you want to understand how a circle works at a deeper level, you will likely need to dig deep in terms of understanding how to identify a circle from an equation.

Example: Calculating the area from a diameter

Assume that the diameter of a circle is \(d = 3\) meters. Find its area.

Solution:The area formula is given in terms of the radius, so the first thing we need is to compute the radius from the diameter. The radius is half of the diameter, so then we get

\[r = \frac{d}{2} = \frac{3}{2} \text{ meters}\]

So, now that we have the radius, we need to use the area formula:

\[\text{Area} = \pi \cdot r^2 = \pi \cdot \left( \displaystyle \frac{3}{2} \right)^2 = \pi \cdot \left( \displaystyle \frac{3^2}{2^2} \right)\] \[ = \pi \cdot \left( \displaystyle \frac{9}{4} \right) = \displaystyle \frac{9\pi }{4}\]

Hence, the area is \( A = \displaystyle \frac{9\pi }{4} \text{ meter}^2\).

This concludes the calculation.

Other useful circle calculators

Circles can be written in different forms, and for example, you can put a circle in standard form, as they could be originally given as a quadratic form in which it is unclear whether it is a circle or not.

There are other things you could do like find the circle form a diameter, or go directly into computing the area and perimeter of a circle.