Concentric Circles

Introduction

Concentric circles, congruent circles & tangent circles are the types of circles. The circle is one of the basic shape in geometry. It is a closed two-dimensional shape formed due to points that are placed equidistant from the central point. The word circle is derived from the Greek word Kirkos it means “hoop” or “ring”. A circle divide a plane interior i.e inside a circle & exterior i.e outside a circle. A circle is measured in the term of its radius. Congruent circles, concentric circles, intersecting circles & tangent circles are different types of circles. In this tutorial, we will study concentric circles. Circles are said to be concentric if they have the same centre & different radii. Let’s study the concentric circle with solved examples.

Circle

• A circle is a two-dimensional closed shape in geometry.

• It can be defined as a set of points which are equidistant from the fixed points in the plane.

• It is a symmetrical shape.

• Congruent circles, concentric circles, tangent circles, semicircles, and intersecting circles are different types of circles.

• The length of the complete circle is known as the circumference of the circle. It is calculated by a formula 2𝜋𝑟.

Basic terminologies regarding circles:

Centre

The fixed point which is used to define a circle is known as the centre of the circle. In the above figure point, O is the centre of a circle.

Radius

The segment joining the centre of the circle to any point of the circle is known as the radius.

Chord

A segment joining any two points of the circle is known as the chord of the circle.

Diameter

A chord which is passed through the centre of the circle is known as the diameter of a circle.

Tangent

A line which touches a circle at a point. It lies outside the circle.

Secant

A line which intersects two points on the circumference is called a secant.

Arc

It is a curved part between any two points of a circle known as an arc.

Segment

A portion between the chord & corresponding arc is known as a segment.

Sector

The area enclosed by two radii & their respective arcs is known as a sector of a circle.

Some properties of the circle:

• Circles are said to be congruent if they have the same radii.

• The diameter of a circle is the longest chord of a circle.

• Circles having different radii are similar.

• A circle having a radius one is known as a unit circle.

• Tangent drawn from an external point to a circle is congruent.

• A circle can be inscribed inside a square, triangle & kite.

Some important formulas of the circle:

Parameter	Formula
Diameter of circle	$\mathrm{D\:=\:2r,\:Where\:r\:is\:radius}$
Area of circle	$\mathrm{A\:=\:\pi\:r^{2}\:or\:\frac{\pi}{4}\times\:d^{2}}$
Circumference	$\mathrm{C\:=\:2\pi\:r\:or\:\pi\:d}$
Area of sector	$\mathrm{A\:=\:\frac{\theta}{360}\times\:2\pi\:r}$
Length of arc	$\mathrm{l\:=\:\frac{\theta}{360}/times\:2\pi\:r}$
Area of segment	$\mathrm{A\:=\:r^{2}[\frac{\pi\:\theta}{360}\:-\:\frac{\sin\:\theta}{360}]}$

(Note: Here 𝜃 represents the measure of arc.)

Concentric circles :

In geometry, objects are said to be concentric if they have a common centre point. If two or more circles with a common centre point & different radii are known as concentric circles.

The figure below shows concentric circles with centre C.

In the above figure, a flat ring-shaped region formed by two concentric circles is known as an annulus.

The word annulus is derived from the Latin word annulus which means ring.

Real-life examples of concentric circles are the wheel of sheep, bull’s eye, and dartboard.

Area of the ring

The above figure shows two concentric circles. The shaded portion represents the annulus or ring. The radius of the outer circle is R & radius of the inner circle is r. The area of the annulus or ring can be calculated by finding the difference between the areas of two circles.

The area of the circle is calculated by using the formula $\mathrm{A\:=\:\pi\:r^{2}}$

$$\mathrm{Area\:of\:outer\:circle\:=\:\pi\:R^{2}}$$

$$\mathrm{Area\:of\:inner\:circle\:=\:\pi\:r^{2}}$$

$$\mathrm{Area\:of\:ring\:=\:Difference\:of\:areas\:of\:two\:circles\:=\:\pi\:R^{2}\:-\:\pi\:r^{2}\:=\:\pi\:(R^{2}\:-\:r^{2})}$$

$$\mathrm{Area\:of\:ring\:=\:\pi\:(R^{2}\:-\:r^{2})}$$

Solved examples

1) Find the annulus of concentric circles if the diameter of two circles are 14 cm & 36 cm respectively.

Given − Diameter of inner circle = 14 cm

∴ Radius of the inner circle (r) = 7cm

Diameter of outer circle = 36 cm

∴ The radius of the outer circle (R) =18 cm

Solution −

$$\mathrm{Annulus\:=\:\pi\:(R^{2}\:-\:r^{2})}$$

$$\mathrm{=\:\frac{22}{7}\times\:(18^{2}\:-\:7^{2})}$$

$$\mathrm{=\:\frac{22}{7}\times\:(\frac{324}{49})}$$

$$\mathrm{=\frac{22}{7}\times\:275}$$

$$\mathrm{=\:864.28\:sq.cm}$$

2) Find the area of the circle if the circumference of the circle is 66 cm.

Ans.

Given − Circumference of circle = 66 cm

Solution −

$$\mathrm{Circumference\:of\:circle\:=\:2\pi\:r}$$

$$\mathrm{66\:=\:2\times\:\frac{22}{7}\times\:r}$$

$$\mathrm{r\:=\:\frac{66\times\:7}{2\times\:}}$$

$$\mathrm{r\:=\:10.5\:cm}$$

$\mathrm{Area\:of\:circle\:=\:\pi\:r^{2}}$

$$\mathrm{=\:\frac{22}{7}\times\:10.5\:\times\:10.5}$$

$$\mathrm{=\:346.5\:sq.cm}$$

3) The diameter of the circular garden is 46 m. There is a 4.5 m road around the garden. Find the area of the road.

Answer: Diameter of circular garden = 46 m

∴ The radius of the inner circle(r) $\mathrm{=\:\frac{46}{2}\:=\:23\:m}$

Width of road = 4.5 m

∴ Radius of outer circle = Radius of inner circle + Width

$$\mathrm{=\:23\:+\:4.5}$$

$$\mathrm{=\:27.5m}$$

A circular garden & road forms a ring, therefore the area of the road is calculated by using a formula,

$$\mathrm{Area\:of\:road\:=\:\pi\:(R^{2}\:-\:r^{2})}$$

$$\mathrm{=\:\frac{22}{7}\times\:({22.5}^{2}\:-\:{23}^{2})}$$

$$\mathrm{=\:\frac{22}{7}\times\:(756.25\:-\:529)}$$

$$\mathrm{=\:\frac{22}{7}\times\:(227.25)}$$

$$\mathrm{=\:714.21\:m^{2}}$$

Therefore the area of road is 714.21 𝑚2

Conclusion

• This tutorial covers a topic in concentric circles in brief with solved examples.

• In this tutorial, we have discussed circles, important formulas of circles, concentric circles & area of the ring of concentric circles.

• A circle is one of the basic two-dimensional shape in geometry, which is formed due to points placed equidistant from the centre.

• Circles are said to be concentric if they have a common centre point but they have different radii.

• Bull’s eye, wheels of the ship, and dartboard are real-life examples of concentric circles.

FAQs

1. State whether the following statement is true or false Circle has infinite chords.

True.

2. What is the equation of a concentric circle?

The equation of the concentric circle is $\mathrm{(x\:-\:h)^{2}\:+\:(y\:-\:k)^{2}\:=\:r^{2}}$

3. )What are the properties of chords?

• The perpendicular drawn from the centre of a circle to its chord bisects the chord.

• The segment joining the centre of a circle & midpoint of its chord is perpendicular to the chord

4. What are three concentric circles?

The inner circle, outer circle & expanding circle are known as three concentric circles.

5. How many circles can be drawn through two given points?

Only one circle can be drawn passing through two given points.