How to find Area of a CircleIn this section, we will learn how to find the area of a circle. At the last of this section, we have explained some examples related to the area of a circle with different scenario for better understanding. Circle: A circle can be defined as a locus (a set of points in a particular position) of points that maintain equidistant from another point called the center. In other words, a circle is a geometrical shape whose distance from the center to the edge is always the same. Every circle has a center. A circle has the following components:
DefinitionThe reign occupied by a circle in the 2D plane is called the area of a circle. In other words, all the space covered by the circle's circumference is called the area of a circle. In the following figure, the colored area represents the area of the circle. Area of Circle FormulaA = πr^{2} Where, A: represents the area of a circle. π: is a constant whose value is 3.1415 or 22/7. r: represents the radius of a circle Derivation of Area of CircleThere are two methods for deriving the area of circle formula:
Using Rectangles Draw a circle and divide it into 16 equal parts. For better understanding, we have filled two colors (red and green) in these parts. as we have shown in figure(a). After dividing, each part of the circle represents a triangle. Arranged these triangles, as we have shown in the figure(b). The resultant figure forms a parallelogramshaped. The area of parallelogramshaped will be equal to the area of a circle because each part has the same area and equal arc length. The green color parts represent half of the circumference, and the other part in red color represents the other half circumference. If the number of parts increased, the parallelogramshape looks like a rectangle whose length is equal to πr and breadth is equal to r. We know that, The area of a rectangle (A) = w * l Where w is the width, and l is the length of the rectangle. From the figure, width (w) = πr and length (l) = r putting these values in the formula, we get area (A) = πr * r Hence, A = πr^{2} Using Triangle Draw a circle with radius r. In this circle, draw some other concentric circles (the circle that has the same center), as shown in figure (a). Now cut the circle up to the center and spread it. The resultant shape will be a triangle, as we have shown in the figure (b). The base and height of the triangle will be equal to the circumference and radius of the circle, respectively. When we calculate the area of a triangle, it gives the area of a circle as a result. We know that: Area of triangle (A) = ½*base*height From the above figure, base = 2πr and height= r Put the value of base and height in the area of triangle formula, A = 1/2 * (2πr) * r Hence, A = πr^{2} ExamplesWhen the diameter is known Example 1: Find the area of a circle if the diameter of a circle is 15 cm. Solution: We have given, diameter (d) = 15 cm. Area =? First, we will find the radius. We know that, d = 2r From the above formula, r = d/2 r = 15/2 = 7.5 We know that, area of circle (A) = πr^{2} A = 3.14 * (7.5*7.5) A = 3.14 * 56.25 A = 176.625 sq cm. When the radius is known Example 2: Find the area of a circle if the radius of a circle is 10 cm. Solution: We have given, radius (r) = 10 cm. Area =? We know that, area of circle (A) = πr^{2} putting the value of r in the above formula, A = 3.14 * (10*10) A = 3.14 * 100 A = 314.00 sq cm. When circumference is known Example 3: Find the area of a circle if the circumference of a circle is 20 cm. Solution: Given, circumference (C) = 20 cm Area =? We know that, Circumference (C) = 2πr From the above formula, 20 = 2* 3.14 * r r = 20 / 2 * 3.14 r = 10 * 3.14 = 31.40 we know that, area of circle (A) = πr^{2} A = 3.14 * (31.40 * 31.40) A = 3.14 * 985.96 A = 3095.9144 sq cm.
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