A farmer is walking around his field, the shape of which is a circle – a shape we are all familiar with, which appears almost everywhere in nature. He starts walking on its boundary from a point, keeps himself on the border, and returns to the point where he began.
He wants to know how much distance he has covered. The only information about the field he is aware of is the radius of the field (the shortest distance from the center of the field to its boundary).
He plans to grow some crops in the field and wants to buy the appropriate amount of pesticides and crops for it. But he needs to know the area of his circular field to buy the things mentioned above. Again, the only piece of information he has is the field's radius. How can he determine the area of the field? These are some fundamental uses of some properties of the circle. Let's explore this intriguing figure and its properties.
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:
A circle is the collection of the set of all points equidistant from a given point.
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.
The radius of a circle is the distance between the center of the circle and any point on 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:
The chord of a circle is a line segment that joins two distinct points on the circle.
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 diameter of a circle is the chord that passes through the center of the 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.
The circumference of a circle is the distance around a circle. It is just another word for the perimeter 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 ratio of the circumference of a circle to the diameter of the circle is known as pi (\(\pi\)).
The radius and circumference of a circle have the following relationship:
$$\pi=\dfrac{C}{2r}$$
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:
$$C=2\pi r$$
Pi is an irrational number, approximately given by \(3.14159265...\) and it never ends. But for the convenience of calculations, it is approximated to \(3.14\) or as the fraction \(\dfrac{22}{7}\).
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.
The area of a circle is the region occupied by a circle in a two-dimensional plane.
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 \((\text{any unit of length})^2\).
More details can be found in our article on Area of Circles.
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.
If two circles intersect at one and only one point, they are said to be 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.
Two or more circles that share the same center are called Concentric Circles.
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.
We say that two circles are congruent if they are the same in every single way, i.e., they are identical.
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!
Find the radius of a circle whose circumference is \(45\text{ cm}\). ( Take \(\pi=3.14\)).
Solution:
Using the formula for the circumference of a circle:
$$C=2\pi r$$
Substituting for \(C\) and \(\pi\) we get,
$$r=\dfrac{C}{2\pi}$$
$$r=\dfrac{45}{2\times 3.14}$$
$$r=7.165\text{ cm}$$
Hence, the radius is \(7.165\text{ cm}\).
The radius of a circular pond is found to be \(20\) meters. Find the circumference of the pond in relevant units. Take \(\pi=3.14\).
Solution:
The radius is given as \(r=20\text{ m}\), plugging it into the formula of circumference, we get
\[\begin{align}C&=2\pi r=\\&=2(3.14)(20)=\\&=125.6\text{ m}\end{align}\]
Therefore, the circumference of the circular pond is \(125.6\) meters.
The perimeter of a circular bowl is measured using a measuring tape and is found to be \(30\text{ cm}\). But after some time, the tape is lost, but the radius of the bowl is yet to be measured. How can we determine the radius of the bowl without the tape? Take \(\pi=3.14\).
Solution:
We can use the formula of the circumference, as it directly relates the radius to the circumference,
$$C=2\pi r$$
Thus, we have
\[\begin{align}r&=\dfrac{C}{2\pi}=\\&=\dfrac{30}{2(3.14}=\\&\approx 4.78\text{ cm}\end{align}\]
Hence, the radius of the bowl is \(4.78\text{ cm}\), rounded to two decimal places.
The radius of a circular table is given by the manufacturer as \(50\text{ cm}\). A tablecloth must be made for it, so its area is required. What is the area of the table?
Solution:
The radius is \(50\text{ cm}\): \(r=50\text{ cm}\).
Using the formula for the area of a circle, we have
\[\begin{align} A&=\pi r^2=\\&=(3.14)(50)^2=\\&=(3.14)(2500)=\\&=7850\text{ cm}\end{align}\]
Hence, the area of the circular table of radius \(50\text{ cm}\) is \(7850\text{ cm}^2\).
Circles - Key takeaways
- A circle is the set of points equidistant from a given fixed point.
- The circumference of a circle is \(2\pi r\) where \(r\) is the radius of the circle.
- The area of a circle with radius \(r\) is \(\pi r^2\).
- The general equation of a circle describing its center explicitly is given by \((x-h)^2+(y-k)^2=r^2\) where \((h, k)\) is the center and \(r\) is the radius.
- There are three types of circles: Tangent Circles, Concentric Circles , and Congruent Circles.
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