Circles

Circle Definitions in Maths

We encounter many types of shapes in nature; the most symmetrical shape we can imagine is a circle. It is defined as follows:

Fig. 1. A circle with centre $O$ and an arbitrary point $P$ on it.

A circle is a part of the conic section, such as a parabola, ellipse, and hyperbola. Imagine an upright cone; if it is sliced in such a way that it is parallel to its base, then the cross-section formed is a circle. Although a circle is mainly known for its symmetrical properties, it is important to remember its definition as a part of the conic section.

Consider a point on the Cartesian plane, which has an x-coordinate, h, and y-coordinate, k, the set of all points equidistant from the given point will form a circle. This fixed point is known as the center of the circle.

Let there be another arbitrary point with coordinates x and y. Let r be the distance between the given point and the arbitrary point. It is known as the radius of the circle.

To understand what radius is, we have to know what diameter is. To understand diameter, we have to define another quantity, known as the Chord of a circle. In rigorous terms, it is defined as follows:

One can indeed construct infinitely many chords on a circle. From the definition of the chord, we can now define the diameter of a circle:

The Equation of a Circle

The general form of the equation of a circle is

$$r=\sqrt{(x-h{)}^2+(y-k{)}^2}$$

Now squaring both sides, we have

$$(x-h{)}^2+(y-k{)}^2=r^2$$

Here $(h,k)$ is the center of the circle and $r$ is the radius.

Circumference of a Circle

Earlier, we saw the farmer going around his field and wanted to measure the distance he had covered. It is nothing but the circumference of a circle.

If you draw a circle and trace it from one point and stop at the same point after one round, the distance you have outlined is the circumference of that circle.

Circumference of a Circle Equation

To find the circumference of a circle, the concept of pi $\pi$ is essential.

Every circle one can draw, at its core, has one property in common. This property or characteristic is what gives rise to pi.

The radius and circumference of a circle have the following relationship:

$$\pi ={\displaystyle \frac{C}{2r}}$$

where $C$ denotes the circumference of the circle and r is its radius. We recall that the diameter is twice the radius. This is how, from the definition of pi, we get the formula for the circumference of a circle:

$$C=2\pi r$$

The Area of a Circle

To help the farmer estimate how many pesticides and crops he will need for his field, we will discuss the area of a circle.

Area of a circle equation

The area of a circle can be derived by cutting the circle into small pieces as follows.

Fig. 2. A circle broke up into pieces to form an approximate rectangle.

If we break the circle into little triangular pieces (like that of a pizza slice) and put them together so that a rectangle is formed, it may not look like an exact rectangle. But if we cut the circle into thin enough slices, we can approximate it to a rectangle.

Observe that we have divided the slices into two equal parts and color them blue and yellow to differentiate them. Hence, the length of the rectangle formed will be half of the circumference of the circle which will be $\pi \times r$. And the breadth will be the size of the slice, which is equal to the circle's radius, $r$.

We did this because we have the formula to calculate the area of a rectangle: the length times the breadth. Thus, we have

$$A=(\pi \times r)\times r$$

$$A=\pi r^2$$

Verbally, the area of a circle with radius $r$ is equal to $\pi$ times the radius squared. Hence, the units of area are ${\text{cm}}^2,{\text{m}}^2$ or $(\text{any unit of length}{)}^2$.

Types of Circles

Circles are of various types, which are uniquely related to each other. Such circles are classified into three types, as follows:

Tangent Circles

Imagine two circles – they need not be congruent and can intersect in infinitely many ways. But a unique way of them crossing will be when they intersect at one and only one point. Such circles are known as Tangent Circles.

They look like this:

Fig. 3. Tangent circles intersect each other at one point.

As seen in the above diagram, two circles intersect at a single point, making them Tangent Circles.

Concentric Circles

The word 'concentric' means ‘having one center,’ which leads to the definition.

Unlike Tangent Circles, where the circles intersect at one and only one point, Concentric Circles have the unique property of never crossing with one another. Concentric circles look like this:

Fig. 4. Concentric circles share the same center point and never intersect.

In terms of the equation of these circles, since the center remains the same, the equations only differ in terms of the radius.

Congruent Circles

Draw a circle and duplicate it, you get two Congruent Circles.

There is not much to be said about congruent circles apart from the fact that they are identical and need not exist at a specific location on a Cartesian plane. Here is a diagram of what two identical circles look like:

Fig. 5. Two congruent circles.

Examples of Circles in Maths

Let's look at a few examples!