# Diameter

 1 Introduction to Circle 2 The Diameter of a Circle 3 Diameter Symbols 4 How do you find a Diameter? 5 Important Notes on Diameter 6 Solved Examples on Diameter 7 Challenging Questions on Diameter 8 Practice Questions on Diameter 9 Maths Olympiad Sample Papers 10 Frequently Asked Questions (FAQs)

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## Introduction to Circle

A circle is a two-dimensional plane figure. It is a collection of points in which all the points are equidistant from the center point  $$\text {O}$$.

Here are some real-life examples of objects which are circular in shape.

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## The Diameter of a Circle

The diameter of a circle is a straight line joining a point from one end of the circle to a point on the other end of the circle, passing through the center.

Look at this image.

Points A, B, M, N, P, Q, X, Y lie on the circle.

'AB' is the line segment which connects the point 'A'  and point 'B'  passing through the center 'O'.

This is the diameter.

Similarly, 'MN', 'PQ', 'XY' also pass through the center  'O'  joining the corresponding points on either side of the circle.

These line segments have the same length.

$$\text{AB = PQ = MN = XY}$$

Note that, 'OA', 'OB', 'OP', 'OQ', 'OM', 'ON', 'OX', 'OY' are half the length of the diameter.

This is called Radius (radii in plural). All radii of the same circle are equal in length.

Radius refers to the line drawn from the center of the circle to an endpoint on the circle.

## Diameter Symbols

Φ is the symbol for diameter.

The diameter of a circle is abbreviated as  'D', 'DIA', 'diam', 'dia',  'd'.

In lower case, diameter symbol is written as φ or ϕ.

The diameter symbol is commonly used in technical specifications and drawings.

A Φ$$25\: \text {mm}$$ means the diameter of the circle is $$25 \: \text {mm}$$.

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## How do you find a Diameter?

Let us understand a few terms before we learn the diameter of a circle formula.

We just learnt that radius ($$\text{r}$$) is the length of the line segment from the center of the circle to an endpoint on the circle.

Circumference ($$\text{C}$$) refers to the enclosed boundary of the circle. We can say that it is the perimeter of the circle.

We can derive the diameter formula from circumference. Let us solve a few example sums here

 $$\text{C= \(\pi$$d}\)

where;

• $$\text C$$ is the circumference
• $$\text d$$ is the diameter
• $$\pi$$ is the constant $$3.14159...$$

The diameter formula when the radius is known is:

 $$\text{Diameter} = 2 \times \text{radius}$$ $$\text {D = 2r}$$

The diameter formula using circumference is:

 $$\text{Diameter} = \frac{\text{Circumference}}{\pi}$$ $\text{D}= \frac{\text{C}}{\pi}$

The diameter formula when the area of the circle is known is :

\begin{align} \text{Area of a circle (A)} &= \pi r^2 \\ \text {Diameter (D)} &= 2\text {r} \end{align}

Simplifying further, we get

 $$D = 2 \sqrt{\frac{ \text {A}}{\pi}}$$

Important Notes
1. $$\ \pi$$ is a mathematical constant which is the ratio of the circumference of a circle to its diameter. It is an irrational number often approximated to $$3.14159$$.
2. Diameter divides the circle into two equal halves called semi-circles.
3. Diameter of a circle formula with reference to Radius (r), Circumference(C) and Area (A) is
* $$\text {D = 2r}$$
* $$\text{D}= \frac{\text{C}}{\pi}$$
* $$D = 2 \sqrt{\frac{ \text {A}}{\pi}}$$

## Solved Examples

 Example 1

The radius of a circle is $$5\: \text{cm}$$. Calculate its diameter.

Solution:

We know that

\begin{align} \text{Diameter} &= 2 \times \text{radius}\\ &= 2\times 5 \end{align}

 $\therefore \text{Radius} = 3 \: \text {cm}$
 Example 2

The diameter of a circle is $$6\: cm$$. What is the radius?

Solution:

We know that

\begin{align} \text{Diameter} &= 2 \times \text{radius}\\ Radius &= \frac{D}{2}\\ &= \frac{6}{2} \\ &=3 \end{align}

 $\therefore \text{Radius} = 3 \: \text {cm}$
 Example 3

The diameter of a circular swimming pool is $$7 \:\text {feet}$$. What is the circumference of the swimming pool? Express your answer in terms of $$\ \pi$$.

Solution:

We know that circumference  a circle

= $$\pi\ \times\text{d}$$

Therefore, Circumference of the swimming pool

=$$\pi \times 7$$

 $\therefore \text{Circumference} = 7 \pi\: \text {feet}$
 Example 4

John walks around a circular park of radius $$100\: \text{m}$$. How many times should he walk around the park to cover at least $$3\: \text{km}$$? (use $$\ \pi\ = 3.14$$)

Solution:

$$=100\: \text{m}$$

Diameter of the park

\begin{align} &= 2\times \text{r}\\ &= 2 \times 100 \\ &=200 \: \text{m} \end{align}

Circumference of the park

\begin{align} &=\pi\ \times \text{diameter}\\ &= \pi \times 200 \\ &= 3.14 \times 200 \\ &= 628 \:\text{m} \end{align}

John covers a distance of $$628 \: \text{m}$$ by going around the park once.

By walking $$4$$ rounds, he covers a distance of $$628\times 4 = 2512 \: \text{m}$$ only.

Instead, he has to go around the park $$5$$ times to cover at least $$3 \:\text{km}$$.

 $\therefore \text{John walks around the park}\: 5 \:\text{times.}$
 Example 5

Jack drew a circle of radius $$3\: cm$$. Tim drew a circle whose radius is $$3$$ times that of Jack's circle. What is the diameter of the circle drawn by Tim?

Solution:

\begin{align} &= 3\times 3 \\ &= 9 \: \text{cm} \end{align}

Diameter  of Tim's circle

\begin{align} &=2 \times radius\\ &= 2\times 9 \\ &= 18\: \text{cm} \end{align}

 $\therefore \text{Diameter of Tim's circle} = 18 \:\text{cm}$

Challenging Questions
1. What is the perimeter of a semi-circle whose radius is $$7\: \text cm$$?
2. A circular wheel has a diameter of $$16 \:\text{cm}$$. How far will it roll if it makes $$10$$ revolutions? (Use $$\pi\ = 3.14$$)
3. If the diameter of a circle is doubled, how does this affect its area?
4. Find the ratio of the circumference of a circle to its area when the radius is $$1\: \text cm$$.

## Practice Questions

Here are a few problems related to diameter.

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

## 1. What is an example of a diameter?

If you observe the wheel of a cycle, the spikes run from one end to another end through the center.

We can relate this to the diameter of a circle, as the diameter is the line segment running from one end to the other end of the circle passing through the center.

## 2. What's the diameter of the circle?

The diameter of a circle is the longest chord of the circle. It is a line segment from one endpoint to another endpoint, passing through the center.

## 3. What is diameter and radius?

Radius refers to the line segment from the center of the circle to an endpoint on the circle. The diameter of a circle is twice the length of the radius, or it can be said that the radius is half the length of the diameter.

$\text{Diameter} = 2 \:\times \: \text{radius}$

More Important Topics
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Algebra
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