A = π r²

Solution

A ≈

# Area of a Circle

The area of a circle can be considered as all the space inside a circle’s circumference. A circle is the curve traced out by a point that moves in such a way that its distance from a given point always constant.

According to Euclid (in Elements book), a circle is a plane figure bounded by one line, such that all right lines drawn from a certain point (called center of the circle) within it to the bounding line, are equal. The bounding line is called its circumference and the point is called its center.

There are some terms need to be understand well before finding the area of a circle.

Circumference: The circumference of a closed curve (circular object) is the linear distance around its edge. The perimeter of a circle or ellipse often called the circumference can be thought of as the length of the outline of a shape. The circumference/perimeter is proportional to its diameter and its radius, that is, there exists a constant number pi, ($\pi$) such that the circle’s perimeter (say P) and its diameter (say D) then $P=\pi D$. In terms of the radius (say R) of the circle then $P=2\pi R$.

Radius: The distance between any of the points and the center point (of the circle) is called the radius of that circle. The radius of a circle is always one-half of the diameter. The radius of the circle with perimeter (circumference)

C is $R=\frac{C}{2\pi}$ or $R=\frac{D}{2}$

Diameter: The diameter of a circle is any straight line segment that passes through the center point of the circle and whose endpoints lie on the circle. The diameter D is the twice of the radius. The diameter can be found using $D=2R$.

$\pi$: The $pi$ is the ratio of the circumference (C) of a circle to its diameter (D). The approximate value of $\pi$ is 3.141592653589 to some decimal places.

Formulas for calculating the area of a circle

The area of any circle can be computed, if the radius or diameter or circumference of the circle is known. Following are the formulas to compute the area of a circle;

1) If you know the value of the radius then the area of a circle is $A = \pi \times R^2$

2) If you know the value of the diameter then area of a circle is $A= \pi \times \frac{D^2}{4}$

3) If you know the value of the circumference then area of a circle is $A=\frac{C^2}{4\pi}$

How to calculate the area of a circle: Some Examples

1) Find the area of a circle is its radius is of 3 meter?

\begin{aligned}

R & = 3 \text{ meter}\\

Area & = pi \times R^2\\

&= \pi \times 3^2\\

&= 3.14159 \times 9 = 28.27 {meter}^2

\end{aligned}

2) What is the area of circle if its diameter is 0.4 meter.

\begin{aligned}

D &=0.4 { meter}\\

Area &= \pi \times \frac{D^2}{4}\\

&=3.14159 \times \frac{0.4}{4}\\

&=0.1256\, {meter}^2

\end{aligned}

3) The area of a circle is 78.5 square inches. What is the radius?

begin{aligned}

R&=78.5\\

A &=3.14159 \times R^2\\

\frac{78.5}{3.14159} &=R^2\\

24.987 &=R^2\\

R = 4.998

\end{aligned}

Note that, the area of a circle is measured in terms of square unit.