A circle is a round plane figure with a boundary that consists of a set of points that are equidistant from a fixed point. This point is known as the center of the circle. There are several measurements associated with the circle. The *circumference* of a circle is essentially the measurement all the way around the figure. It is the enclosing boundary, or the edge. The *radius* of a circle is a straight line segment from the circle's center point to the outer edge. This can be measured using the center point of the circle and any point on the edge of circle as its end points. The *diameter* of a circle is the straight-line measurement from one edge of the circle to the other, crossing through the center.

The *surface area* of a circle, or any two-dimensional closed curve, is the total area contained by that curve. The area of a circle may be calculated when the length of its radius, diameter, or circumference is known.

#### TL;DR (Too Long; Didn't Read)

The formula for the surface area of a circle is *A* = π_r_^{2}, where *A* is the area of the circle and *r* is the radius of the circle.

## An Introduction to Pi

In order to calculate the area of a circle you'll need to understand the concept of Pi. Pi, represented in math problems by π (the sixteenth letter of the Greek alphabet), is defined as the ratio of a circle's circumference to its diameter. It is a constant ratio of the circumference to the diameter. This means that π = *c*/*d,* where c is the circumference of a circle and *d* is the diameter of the same circle.

The exact value of π can never be known, but it can be estimated to any desired accuracy. The value of π to six decimal places is 3.141593. However, the decimal places of π go on and on without a specific pattern or end, so for most applications the value of π is customarily abbreviated to 3.14, especially when calculating with pencil and paper.

## The Area of a Circle Formula

Examine the "area of a circle" formula: *A* = π_r_^{2}, where *A* is the area of the circle and *r* is the radius of the circle. Archimedes proved this in approximately 260 B.C. using the law of contradiction, and modern mathematics does so more rigorously with integral calculus.

## Apply the Surface Area Formula

Now it's time to use the formula just discussed to calculate the area of a circle with a known radius. Imagine that you're asked to find the area of a circle with a radius of 2.

The formula for the area of that circle is *A* = π_r_^{2}.

Substituting the known value of *r* into the equation gives you *A =* π(2^{2}) = π(4).

Substituting the accepted value of 3.14 for π, you have *A* = 4 × 3.14, or approximately 12.57.

## Formula for Area From Diameter

You can convert the formula for area of a circle to calculate area using the circle's diameter, *d*. Since 2_r_ = *d* is an unequal equation, both sides of the equal sign must be balanced. If you divide each side by 2, the result will be *r* = _d/_2. Substituting this into the general formula for area of a circle, you have:

*A* = π_r_^{2} = π(*d*/2)^{2} = π(d^{2})/4.

## Formula for Area From Circumference

You can also convert the original equation to calculate the area of a circle from its circumference, *c*. We know that π = *c*/*d*; rewriting this in terms of *d* you have *d* = *c*/π.

Substituting this value for *d* into *A* = π(*d*^{2})/4, we have the modified formula:

*A* = π((*c*/π)^{2})/4 = *c*^{2}/(4 × π).