# 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 th**e** 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 2** π**r(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