How to Calculate the Area of a Circle from Its Circumference
The most commonly taught formula for the area of a circle uses the radius or diameter. If the radius is R and the diameter is D = 2R, then the area of the circle is given by the equations
area = πR^2
area = (π/4)D^2
But to measure the radius or diameter of a circle, you must know where its center is. The radius is the distance from the center to the edge of the circle, and the diameter is a the distance across the circle passing through the center. Without any idea of where the center is, you can't accurately determine the radius or diameter, and therefore you can't use one of the formulas above.
However, it is possible to find the area of a circle using the length of the boundary, the circumference. The circumference is easy to measure because you can start from any point on the circle's edge and work you way around.
Area Formula in Terms of Circumference
If the circumference of the circle is C, then the area is given by the equation
area = (C^2)/(4π)
This geometry formula has the mathematical constant π in the denominator of a fraction, rather than the numerator where we normally find it. In words, the formula means that you multiply the circumference by itself, then divide that product by 4π. The value of 4π is approximately 12.5664.
How did we come up with this formula?
In terms of the radius, the circumference is given by C = 2πR. Solving this for R, we see that R = C/(2π). If we plug this expression into the very first area formula, we get
area = πR^2
= π[ C/(2π) ]^2
= π[ (C^2) / (4π^2) ]
= (C^2) / (4π)
When would we ever need to use this formula for the area of a circle? Suppose you have a large circular pool that is 1.345 meters deep, but you don't know its diameter or radius. If you want to accurately compute the capacity of the pool, you first measure around the perimeter of the pool with a flexible measuring tape to determine the circumference.
Say the circumference is 29.783 meters. Plugging the value C = 29.783 into the expression (C^2)/(4π) we find that the area of the base of the pool is equal to
(29.783^2)/(4π) = 70.587373 square meters.
The volume of the pool is the area of the base times the height, so you obtain
1.345*70.587373 = 94.940016685 cubic meters
In liters, the capacity of the pool is about 94940.017 L.
More Geometry Guides
You have 25 feet of flexible material to fence off an enclosed area. You know that given any fixed length of material, the shape that yields the maximum area is a circle. What is the largest area you can enclose with 25 feet of fencing?
For this problem, we have C = 25 feet. Plugging this value of C into the area formula we get
area = (C^2)/(4π)
= 49.7359 square feet.
The simplified version of the area formula is area ≈ 0.07958*C*C. All you have to remember is to multiply the constant 0.07958 by the circumference squared. Let's use this simplified formula to estimate the area of a region that can be approximated by a circle.
A landowner has 45 pieces of rigid fencing material to build an enclosure. Each piece of fence is 7 feet long. He constructs an enclosed area in the shape of a regular 45-sided polygon by placing the fence pieces end to end at an angle of 172°. What is the approximate area of the enclosed region?
The more sides a regular polygon has, the more it resembles a circle. Thus, the 45-sided regular polygon can be approximated to a circle with a circumference of 45*7 = 315 feet. This means we have C = 315. Plugging it into the simplified formula gives us
area ≈ 0.07958*315*315
= 7896.3255 square feet.
So the area enclosed by the fence is approximately 7896 square feet. Using the formula for the area of a regular polygon, the exact area is about 7883.2423, so the estimation gives a slight overestimate by 0.16%.
Circumference from Area
Using the fact that area = (C^2)/(4π), we can solve for the circumference given the area. Call the area A, then we have
A = (C^2)/(4π)
4πA = C^2
sqrt(4πA) = C
For example, if the area of a circle is 38.45 cm^2, then the circumference is
C = sqrt(4π*38.45)
= 21.98 cm.