Math Doubts

Geometric proof of Area of an Annulus

The formula for finding the area of an annulus (or a circular ring) is written as follows.

$A$ $\,=\,$ $\pi(R^2-r^2)$

It can be derived in mathematical form geometrically. Let’s learn how to find the area of a circular ring (or an annulus) by the geometrical approach.

Areas of the circles

circles

Firstly, consider two circles and their radii are different. The radius of one circle is denoted by $R$ and the radius of the second circle is represented by $r$. Here, $R \,>\, r$

Similarly, the areas of both circles are denoted $A_1$ and $A_2$ respectively. According to the area of a circle formula, the area of each circle can be written as follows.

$(1).\,\,\,$ $A_1 \,=\, \pi R^2$

$(2).\,\,\,$ $A_2 \,=\, \pi r^2$

Area of the concentric circles

The two circles are arranged in such a way that the centers of them coincide at a point. The concentric circles have a common centre and their subtraction forms a geometric shape, called an annulus or a circular ring. Let’s denote its area by $A$.

area of an annulus

The area of circular ring can be obtained by subtracting the area of the small circle from the area of the big circle.

$A$ $\,=\,$ $A_1-A_2$

Now, substitute areas of both circles in mathematical form in the above equation.

$\implies$ $A$ $\,=\,$ $\pi R^2-\pi r^2$

$\implies$ $A$ $\,=\,$ $\pi \times R^2-\pi \times r^2$

Pi is a common factor in both terms on the right hand side of the equation. It can be taken out common from them for simplifying the expression further.

$\implies$ $A$ $\,=\,$ $\pi \times (R^2-r^2)$

$\,\,\,\therefore\,\,\,\,\,\,$ $A$ $\,=\,$ $\pi(R^2-r^2)$

It is a formula to find the area of an annulus or a circular ring from the radii of the two concentric circles.

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