Area of Circle

A circle is one of the most prominent two dimensional shapes. Online Geometry Tutoring provides you all the help for your homework. The surface covered by the interior of this shape is called as the area of circle. Like we have formulas for areas, we do have a formula for the area of a circle. Let us see what is the formula for the area of a circle. It is very simple unlike we have formulas in organic chemistry like cyanide formula. The formula for the area of a circle is A = πr^2, where ‘r’ is the radius of the circle and π is a constant, an irrational number. The algebraic approximation of its value can reasonably considered as either 3.14 in decimals or as 22/7 in fraction.

The above formula cannot be derived that easily as we do in case of areas of squares or rectangles where we use the concept of unit squares. However, applying the Concept Of Calculus it can be derived easily. In this article let us not concentrate on the derivation of the formula and accept that as it is.

But the question arises how to calculate the area of a circle in different situations. For example, we will illustrate the following case. The diagram below shows a circle that is inscribed in a scalene triangle and we know only the measures of the sides of the triangle.

In the above diagram ABC is a scalene triangle and a circle with center as O and radius as ‘r’ is inscribed. In other words, the sides of the triangle are tangents to the inscribed circle. The circle is also called as in-circle and its center is called as the in-center of the triangle. The measures of the sides are ‘a’, ‘b’ and ‘c’ corresponding to the angles A, B and C at the vertices. The area of the triangle can be given as per Heron’s formula as A = √[(s(s – a)(s – b)(s – c)], where’s’ is half the perimeter of the triangle ABC.

At the same time, the area of the triangle ABC can also be computed as,
A = area of triangle BOC + area of triangle COA + are of triangle AOB. As per the property of in-center of a triangle, the perpendiculars OD, OE and OF from O to the sides BC, CA and AB are all congruent and equal to the radius ‘r’ of the in-circle.

So,
A = (½)a*r + (½)b*r + (½)c*r = [(a + b + c)/2]*r = s*r, since ‘s’ denotes half the perimeter of the triangle. Therefore, equating both the above results,
s*r = √[(s(s – a)(s – b)(s – c)] or r = √[((s – a)(s – b)(s – c)/(s)]

Thus, if the measures of the sides of a triangle are known, the measure ‘r’ of the radius of the in-circle can be figured out and subsequently the area of the circle that is inscribed can be determined from the formula A = πr^2.