Diameter Formula: Definition, Formulas and Solved Examples

The Diameter of a circle is a straight line that touches both sides of a circle and passes through the centre. Some of the round objects are bangles, coins, clock dials, etc. In three dimensional geometry, these round figures are known as spheres, and in two-dimensional geometry, these are known as circles.

The distance from the points on the circumference or boundary of a circle to its centre is known as the circle’s radius. Diameter is the line segment joining two points on the boundary or circumference of a circle that passes through the centre. In this article, we will learn about diameters and formulas associated with the diameter of a circle.

Diameter: Definition

In a plane, if a point moves in such a way that its distance always remains the same from a fixed point, then the locus of the point is called a circle. The fixed point is called the centre of the circle. Circles are objects that are round in shape. Some of the round objects are bangles, coins, dials of the clock etc.

Some important terms related to circles are discussed below, which helps solve and understand the concepts of the circle’s diameter easily.

1. Centre: The fixed or centre point of the circle.

2. Radius: The distance from the points on the circumference or boundary of the circle to the centre of a circle is known as the radius of the circle.

3. Chord: The line segment connecting any two points that lie on the boundary or circumference of the circle.

4. Circumference: The length of the boundary of the circle is called its circumference or perimeter.

5. Area: In a two-dimensional plane, the amount of region or space enclosed by the circle is called its area.

6. Diameter: Diameter is the line segment connecting two points on the circumference of a circle that passes through the centre of the circle.

Learn About Circumference of a Circle

In general, the numerical length of the circle’s diameter equals twice or double the radius of the given circle. In the above figure, the line segment \(MN\) gives the diameter of the circle. There are infinite diameters that can be drawn in any given circle. The diameter is said to be the longest chord of the circle. The diameter divides the given circle into two equal parts (semi-circle).

Diameter Formula of a Circle

The diameter formula is the one used to calculate the diameter of a circle. The diameter of the circle can be calculated using any of the information given below:
1. Radius of the circle
2. Circumference or perimeter of the circle
3. Area of the circle
By knowing any of the above, we can find or formulate the diameter of the circle.

Diameter \(\left( d \right)\) formulas in various situations are given below:

(i). Diameter \(\left( d \right) = 2 \times {\rm{radius}}\,\left( r \right)\)
(ii). Diameter \(\left( d \right) = \frac{{{\rm{circumference}}}}{\pi }\)
(iii). Diameter \(\left( d \right) = 2\sqrt {\frac{{{\rm{area}}}}{\pi }} \)
Here, \(\pi \) is constant, and its general value is \(\frac{{22}}{7}\) or \(3.14\) (Approx).

Finding Diameter of a Circle Using Radius

In the above figure, \(OM,\,ON\) are the radius of the circle and \(OM = ON = r.\)
And, the line segment \(MN = OM + ON = 2\,OM = 2\,r\) is known as the diameter of the circle.
Thus, the diameter of the circle equals twice the radius of the circle.
\(d = 2\,r\)
Here, \(d\) is the diameter of the circle and \(r\) is the radius of the circle.

Diameter of a Circle from Circumference: Method

We know that the length of the boundary of the circle is called its circumference or perimeter.

Let us consider the circle of radius \(“r”,\) then the circle’s circumference is given by \(2\pi r.\)
\(C = 2\pi r\)
But we know that, diameter \(\left( d \right) = 2 \times {\rm{radius}}\,\left( r \right)\)
So, circumference \(\left( C \right) = \pi d\)
From the above,
\(d = \frac{{{\rm{circumference}}}}{\pi }\)
Here, \(\pi \) is constant, and its general value is \(\frac{{22}}{7}\) or \(3.14\) (Approx).

Diameter of a Circle With Area: Method

In a two-dimensional plane, the amount of region or space enclosed by the circle is called the circle area.

The area of the circle with radius \(“r”\) is given by \(\pi {r^2}.\)

Area \(\left( A \right) = \pi {r^2}\)
Multiply and divide the above equation by \(4.\)
\( \Rightarrow {\rm{Area}} = \frac{{4\pi {r^2}}}{4} = \frac{{\pi {{\left( {2r} \right)}^2}}}{4}\)
But we know that, diameter \(\left( d \right) = 2 \times {\rm{radius}}\,\left( r \right)\)
\( \Rightarrow {\rm{Area}} = \frac{{\pi {d^2}}}{4}\)
\( \Rightarrow {d^2} = 4 \times \frac{{{\rm{Area}}}}{\pi }\)
\(d = 2\sqrt {\frac{{{\rm{Area}}}}{\pi }} \)
Here, \(\pi – \)constant and its general value is \(\frac{{22}}{7}\) or \(3.14\) (Approx).

Diameter Formula: Uses

Diameter is the segment of a circle that starts at any point on its circumference, then passes through its center and ends on the circumference at the other side. The diameter of the circle is the longest chord of a circle.

Some of the uses of the diameter formula are discussed below:
1. The diameter formula is used to calculate the radius of the circle or circular base of the solid.
2. The diameter formula is used to calculate the circumference of the circle. Circumference \(= \pi \times {\rm{diameter}}\)
3. The diameter formula is used to calculate the area of the circle.
\({\rm{Area}} = \frac{{\pi {{\left( {{\rm{diameter}}} \right)}^2}}}{4}\)
4. The diameter formula is used to calculate the surface areas of the solids like a cylinder, sphere, cone, hemisphere etc.
5. The diameter formula is used to estimate the boundary of the field.

Chord and Diameter of a Circle

The chord is the line segment joining any two points on the boundary or circumference of the circle. Diameter is the line segment connecting two points that lie on the circumference and passes through the centre of the circle.

Thus, if a chord is passing through the centre of a circle, then it is known as the diameter of the circle. We can also say that the diameter of the circle is the longest chord of the circle.

A circle can have an infinite number of diameters.

Solved Examples – Diameter Formula

Let us look at some of the solved examples:

Q.1. Charan drew the circle with a radius of 3 units. Find the diameter of the circle drawn by the Charan?
Ans:
Given Charan drew the circle with a radius of \(3\) units.

Given \(r = 3\) units.
We know that numerically the diameter of the circle equals twice or double the length of the radius.
Diameter \(\left( d \right) = 2 \times {\rm{radius}}\left( r \right)\)
\(d = 2r = 2 \times 3 = 6\) units.
Hence, the diameter of the circle drawn by the Charan is \(6\) units.

Q.2. The circumference of the swimming pool is given by 7π units. Find the diameter and radius of the swimming pool, which is in the shape of a circle.
Ans: Given swimming pool is in the shape of a circle, and given the circumference of the swimming pool is given by \(7\pi \) units.
We know that circumference of the circle with the diameter \(“d”\) is given by \(\pi d.\)
\( \Rightarrow \pi d = 7\pi \)
\( \Rightarrow d = 7\) units
We know that radius of the circle is half of the radius of the circle.
\(r = \frac{d}{2} = \frac{7}{2} = 3.5\) units
Hence, the diameter and radius of the circle are \(7\) and \(\frac{7}{2}\) units respectively.

Q.3. Find the diameter of the circle whose area is \(300\,{\rm{sq}}.{\rm{cm}}.\) (Use \(\pi = 3.14\))
Ans:
Given the area of the circle is \(300\,{\rm{sq}}.{\rm{cm}}{\rm{.}}\)
Let the diameter of the circle be \(“d”\)
We know that the relation between the area of the circle and the diameter of the circle is given by
\(d = 2\sqrt {\frac{{{\rm{area}}}}{\pi }} \)
\(\Rightarrow d = 2 \times \sqrt {\frac{{300}}{{2.14}}} \)
\( \Rightarrow d = 2 \times 9.77\,{\rm{cm}}\)
\( \Rightarrow d = 19.54\,{\rm{cm}}\)
Hence, the diameter of the circle is \(19.54\,{\rm{cm}}\) (approx..)

Q.4. Find the diameter of the circle, whose area is \(154\,{\rm{c}}{{\rm{m}}^2}.\) Also, find the circumference of the circle by using the diameter formula. (Use: \(\pi = \frac{{22}}{7}\))
Ans:
Given the area of the circle is \(154\,{\rm{c}}{{\rm{m}}^2}.\)
We know that the relation between the area of the circle and the diameter of the circle is given by
\(d = 2\sqrt {\frac{{{\rm{area}}}}{\pi }} \)
\( \Rightarrow d = 2 \times \sqrt {\frac{{154}}{{\frac{{22}}{7}}}} \)
\( \Rightarrow d = 2 \times 7\,{\rm{cm}}\)
\( \Rightarrow d = 14\,{\rm{cm}}\)
The diameter of the circle is \(14\,{\rm{cm}}.\)
We know that the circumference of the circle with diameter \(“d”\) is given by \(\pi d.\)
Circumference \( = \frac{{22}}{7} \times 7 = 22\,{\rm{cm}}\)
Hence, the circumference of the circle is \(22\,{\rm{cm}}{\rm{.}}\)

Q.5. The radius of a circle is \(8\,{\rm{cm}}.\) Find the length of the longest chord drawn to the circle.
Ans: Given the radius of the circle \(\left( r \right) = 8\,{\rm{cm}}.\)
We know that the longest chord of the circle is the diameter of the circle, whose value is given by \(\left( d \right) = 2r.\)
So, the length of the longest chord is \(2 \times 8 = 16\,{\rm{cm}}.\)

Summary

In this article, we have studied the definitions of the circle and the terms related to circles such as radius, diameter, chord, circumference, and circle area. This article gives the diameter formula and uses of diameter formula. This article also discussed finding the diameter of a circle using radius or circumference or the area of the circle. This article includes numerical examples, which helps us understand the concepts and formulas easily.

Learn About the Different Parts of a Circle

Frequently Asked Questions (FAQs) – Diameter Formula

Frequently asked questions related to diameter formula is listed as follows:

Q.1.How do you find the diameter in Math?
Ans: In mathematics, diameter is found by using three ways by using radius or circumference or the area of a circle. These are mentioned below:
1. Diameter \(\left( d \right) = 2 \times {\rm{radius}}\,\left( r \right)\)
2. Diameter \(\left( d \right) = \frac{{{\rm{circumference}}}}{\pi }\)
3. Diameter \(\left( d \right) = 2\sqrt {\frac{{{\rm{area}}}}{\pi }} \)

Q.2. How do you relate the radius and the diameter of a circle?
Ans: We know that numerically the length of the diameter is twice or double the radius of the circle. Thus, the radius and the diameter of a circle are related by the formula:
Diameter \(\left( d \right) = 2 \times {\rm{radius}}\left( r \right).\)

Q.3. How do you find the diameter from the circumference of a circle?
Ans: The diameter of the circle can be found by using the circumference of a circle. It is given by Diameter \(\left( d \right) = \frac{{{\rm{circumference}}}}{\pi }.\)

Q.4. How do you find the diameter from the area of a circle?
Ans: The diameter of the circle can be found by using the area of a circle. It is given by diameter \(\left( d \right) = 2\sqrt {\frac{{{\rm{area}}}}{\pi }} .\)

Q.5. What is the diameter?
Ans: Diameter is the line segment connecting two points that lie on the circumference and passing through the centre of a circle.

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