A circle is one of the most widely recognizable geometric shapes, but exploring the mathematical concepts of diameter and area can sometimes feel tricky. Whether you are measuring the size of round rug you need to purchase or determining the space you need to construct a round garden or patio, knowing how to calculate the area of a circle from its diameter is a valuable skill.
TL;DR (Too Long; Didn't Read)
The area of a circle is the amount of space the circle covers. The formula for calculating the area of a circle is A = πr2 where pi (π) equals 3.14 and the radius (r) is half the diameter.
The first step for calculating the area of a circle from its diameter is to find that diameter. While math problems often list this value, in the real world, you must find the diameter yourself. The diameter is the length of a line that begins at the edge of the circle, passes through the center of the circle, and ends at the opposite edge of the circle. To measure, you will need a ruler for small circles or a tape measure for large circles.
Once you have the diameter (d) of the circle, you can find the radius (r) using the equation d=2r. The radius of a circle is the distance from the center of the circle to any point on the edge of the circle. The radius is also half of the diameter. If your diameter is a simple number, you can likely calculate the radius in your head. If not, rearrange the equation to find for r
r = \frac{d}{2}
and solve.
You are now ready to use the equation for area:
A = πr^2
Pi (π) is a non-algebraic number that represents the ratio of the distance around the circle (circumference) to its diameter, usually estimated as 3.14. To solve for area, square the radius (radius times radius) then multiply by 3.14.
Since area is a measure of two dimensions, you always report area in square units like square inches (in2) or square feet (ft2). This is especially important when calculating the area of a circle for an assignment since an answer without correctly reported units is likely incorrect or incomplete.
Any time you need to determine the space inside a circle or the amount of space a circle covers, you can use the equation for the area of a circle. Especially for real world applications of this skill, measuring diameter is often the simplest way to start.
Use this calculator to easily calculate the area of a circle, given its radius in any metric: mm, cm, meters, km, inches, feet, yards, miles, etc.
Quick navigation:
- Area of a circle formula
- How to calculate the area of a circle?
- Example: find the area of a circle
- Practical application
Area of a circle formula
The formula for the area of a circle is π x radius2, but the diameter of the circle is d = 2 x r2, so another way to write it is π x (diameter / 2)2. Visual on the figure below:
π is, of course, the famous mathematical constant, equal to about 3.14159, which was originally defined as the ratio of a circle's circumference to its diameter. The formula above is the one used in our area of a circle calculator.
How to calculate the area of a circle?
Calculation is easy once you have measured the circle's radius or diameter, or if you know it from plans and schematics: just plug the numbers into the formulas above or use our circle area calculator instead. If you are measuring it by hand, remember that the diameter is the largest measurement you can get from a circle.
Example: find the area of a circle
Task 1: Given the radius of a circle, find its area. For example, if the radius is 5 inches, then using the first area formula calculate π x 52 = 3.14159 x 25 = 78.54 sq in.
Task 2: Find the area of a circle given its diameter is 12 cm. Apply the second equation to get π x (12 / 2)2 = 3.14159 x 36 = 113.1 cm2 (square centimeters).
Practical application
Circle geometry has a wide array of practical uses. Circles are used when planning athletic tracks, recreational areas, buildings, and roundabouts, so knowing their area is important in construction, landscaping, etc. The famous Ferris-wheel attraction is a circle, as are the wheels on your car or bike. Circle-like parts, e.g. cylinders, tubes, gears, and others are used by engineers in clocks, bikes, cars, trains, ships, planes, and even rockets. The invention of the wheel was one of the transforming events in early human history, as it dramatically reduced the energy expended in moving stuff around and made travelling easier. The number π has application in calculating statistical distributions like the normal distribution (gaussian distribution), which are used throughout the sciences.