What Is A Circle?
We can simply say that area of the circle is πr². A circle is a round-shaped figure that has no corners or edges. Which is one of the famous curves in a conic section
Definition of circle
A circle is a shape with all points the same distance from a Point. This point is named the centre, or a circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape
Euclid’s definition of the circle
A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre
How to find the area of circle?
Using geometry
Above the figure, we divide the circle into 8 triangles. Here area of the circle is almost equal to the area of 8 triangles
If we divide the circle into more(infinite) triangles then the sum of the area of triangles = the Area of the circle, in this case, the value of AB is almost zero but not zero. Also, the height of the triangle becomes the radius of the circle
Let assume we can draw y triangles like this inside a circle. Then y×r = 2πr (perimeter of the circle) (r is the radius of the circle). So Area of y triangles = Area of the circle
Area of one triangle = ½ AB×r
Area of y triangles = Area of one triangle × y = ½AB×r×y = ½(2πr)r = πr²
That is Area = πr²
Using integration
From the geometry method, we saw that AB is tenting to zero so let’s consider AB = dx, so the area of one triangle = ½(dx)r. Here we need to integrate ½(dx)r from zero to 2πr(perimeter of the circle).
That is Area of circle = 0∫2πr ½(dx)r = ½r × 0∫2πr dx = ½r[2πr] = πr²
Sample problems : Find the area of circle
1) If the diameter of the circle is 10 cm then What is the area of the circle?
Diameter (D) = 2×Radius(r) = 10 cm
Area = πr² = π(½D)² = π(5)² = 25π cm²
2) If the perimeter is 2π cm then find the area?
Perimeter = 2π × Radius(r) = 2π cm
That is r = 1 cm
Area = πr² = π(1)² = π cm²