Radius Formula: Meaning, Calculation in Different Cases & Solved Example

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Radius of a Circle is a straight line segment that is drawn from the centre of the circle to its outer edges. In simpler terms, a radius can be drawn in any direction from the central point of the circle. R or r is a common acronym and mathematical variable name for radius. Since there are infinite points on the circumference of a circle and all the radii of the circle are equidistant from the centre, an infinite number of radii can be drawn on a circle. We can easily calculate the radius of a circle by dividing the diameter of the circle by 2.

Table of Content

What is Radius?
Finding the Radius of a Circle
Radius Formulas
Things to Remember
Sample Questions

Key Terms: Radius, Diameter, Circumference, Area, Perimeter, Area of a Circle, Radii, Chord, Sphere, Circle, Line Segment

Read More: Arc Length Formula

What is Radius?

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The radius is a line segment that connects the circle's or sphere's centre to its perimeter or boundary. It is commonly abbreviated as 'r' and is an integral part of circles and spheres. The plural of radius is "radii," which is used when discussing multiple radii at once. The diameter is the longest line segment of a circle or sphere that connects any points on the opposite side of the centre, while the radius is half the diameter's length. It can be written as d/2, where d' is the circle or sphere's diameter.

Radius of a Circle

Radius of a Circle

The video below explains this:

Radius Formula Detailed Video Explanation:

Read More: Diameter Formula

Finding the Radius of a Circle

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When the diameter, area, or circumference of a circle is known, the radius can be calculated using the three basic radius formulas which are as follows:

Radius = Diameter/ 2 when the diameter is given.
Radius = Circumference/2π, when the circumference is given.
Radius = \(\sqrt(\)Area/\( \pi \)), when the area of the circle is given.

The value of \(\pi \) is 3.16 or 22/7 for the above-mentioned formulas.

Read More: Perimeter and Area of a Circle

Radius Formulas

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The radius of a circle and sphere may be computed using various formulas that are given in the section below.

Finding Radius of a Circle When Diameter is Given

A straight line passing through the centre and connecting a point on one end to a point on the other end of the circle is called the diameter. The diameter is twice the length of the radius. It is also the circle's longest chord.

The radius formula when the diameter of a circle is known is

Radius = Diameter/2 or D/2 units

\(Radius = \sqrt{\frac{Diametre}{2}}\)

Radius VS Diameter

Radius VS Diameter

Read More: Area of a Sector

Finding Radius of a Circle When Circumference is Given

The circumference of a circle is its perimeter. It is the circle's boundary and may be calculated using the formula C = 2\(\pi\)r units. The circumference of the circle is C, the radius of the circle is r, and the constant is 3.14159. When the circumference is given, the radius is calculated as

Radius = Circumference/2\(\pi\)or C/2\(\pi\)units

\(Radius = \frac{Circumference}{2\pi}\)

Finding Radius of a Circle When Area is Given

The area of a circle is the amount of space it takes up. The formula of the area of the circle is π × radius². When the area is given, the radius can be calculated using the given formula

Radius = \(\sqrt (\)Area/\(\pi\)) units

\(Radius = \sqrt{\frac{Area}{\pi}}\)

Circle and Related Formulas

Circle and Related Formulas

Read More: Areas Related to Circles

Things to Remember

The radius of a circle is the length of a line segment connecting the circle's centre to a point on its perimeter. It is denoted as ‘r’.
The diameter of a circle is double the radius, or, the radius is half the diameter. The relation between radius and diameter can be denoted as Diameter = 2 × radius.
The radius of a circle can be calculated by different formulas.
When the diameter is given, the formula is Radius = Diameter / 2.
When the circumference is given, the formula is Circumference / 2\(\pi\).
When the area is known, the formula is Radius = \(\sqrt (\)Area of the circle /\(\pi\)).
One can also find the area of a circle by using the circumference of the circle. The radius of the circle can be calculated from the circumference of the circle and this value can be used to find the area of the circle.

Read More:

Related Links
Area of Segment of a Circle	The Sector of a Circle	Areas Related To Circles Formula
Geometry Formula	Difference Between Area and Perimeter	Perimeter of a Square
Sphere Formula	Semicircles	Surface Area Of A Hemisphere

Solved Questions

Ques. Find the radius of a circle that has a circumference of 15 inches? (2 Marks)

Ans. We can calculate by using the formula, r = C/2π.

r = 15/2\(\pi\)

r = (15×7)/(2×22)

r = 105/44

r = 2.39 inches

Ques. What is the radius of the circle if the area is 36m²? (2 Marks)

Ans. Area of the circle (A) = 36m²

Formula, r = √(A/\(\pi\))

r = \(?(36/) m\)

r = √[(36×7)/ (22)] m

r = \(\sqrt{11.45 m}\)

r = 3.39 m

Ques. Calculate what will be the radius if the length of the chord of a circle is 8 cm and the perpendicular distance from the centre to the chord is 3 cm. (3 Marks)

Ans. According to the given question,

Length of the chord (AB) = 8 cm

Perpendicular distance = OP = 3 cm

Radius = OA

It is a known fact that the perpendicular line drawn from the centre of a circle to a chord bisects the chord.

AP = PB = 4 cm

In triangle OPA,

Applying Pythagoras theorem,

OA² = OP² + AP²

OA² = 9 + 16

OA² = 25

OA = 5

Therefore, radius = 5 cm

Ques. What is the relationship between the radius and diameter of a circle? (1 Mark)

Ans. The radius is equal to half of the diameter of the circle. So, If the diameter is 20 cm, the radius will be 10 cm.

Ques. Determine the circumference and the area of a circle whose radius is 14 cm. (3 Marks)

Ans. The radius of the circle = 14 cm

Circumference of the Circle = 2πr

= 2 × 22/7 × 14

= 2 × 22 × 2

= 88 cm

Using area of Circle formula, \(\pi r^2\)

= 22/7 × 14 × 14

= 22 × 2 × 14

= 616 sq. cm.

Ques. The ratio of the area of two circles is 4:9. By using the area of circle formula find the ratio of their radii. (3 Marks)

Ans. Suppose,

Radius of the 1st circle = R₁

Area of the 1st circle = A₁

Radius of the 2nd circle = R₂

Area of the 2nd circle = A₂

Given, A₁:A₂ = 4:9

Area of a Circle = \(\pi r^2\)

\(\pi R1^2\): \(\pi R 2^2\) = 4: 9

Taking square roots of both sides,

R₁: R₂ = 2: 3

So, the ratio of the radii = 2:3

Ques. A race track is in the form of a circular ring. The inner radius of the track is 58 yards and the outer radius is 63 yards. Calculate the area of the race track. (3 Marks)

Ans. R (Outer radius) = 63 yards, r (Inner radius) = 56 yds.

Let the area of the outer circle be assumed as A1 and the area of the inner circle be A₂

Area of race track = A₁ - A₂

=\(\pi R^2\) -\(\pi r^2\)

= \(\pi\) (63² - 56²)

= 22/7 × 833

= 2,618 square yards

Ques. To cover a distance of 10 km, a wheel rotates 5000 times. Find the radius of the wheel. (5 Marks)

Ans. Number of rotations = 5000.

Total distance covered = 10 km

Let us assume ‘r’ be the radius of the wheel.

So, the circumference of the wheel = distance covered in 1 rotation = \(2\pi r\).

In 5000 rotations, the distance covered = 10 km = 1000000 cm

Therefore, in 1 rotation, the distance covered = 1000000/5000cm=200cm

However, this is equal to the circumference.

Therefore, \(2\pi r\) = 200 cm

r = 200/\(2\pi \)

r = 100/\(\pi \)

Putting the value of π as 22/7, we get

r = 100 x 7/22

r = 31.82 cm

Ques. The diameter of a semi-circular shape is 14 cm. Find the perimeter. (3 Marks)

Ans. Diameter of semicircle = d = 14 cm

Radius = r = d/2

= 14/2 = 7 cm

Perimeter of semicircle is given as (Perimeter of circle/2) + d

= (\(2\pi r\)/2) + d

= \(\pi r\) + d

= (22/7) × 7 + 14

= 22 + 14

= 36 cm

Ques. The difference between the circumference and the diameter of a circular bangle is given as 5 cm. Determine the radius of the bangle. (Take \(\pi \)=22/7) (3 Marks)

Ans. Let the radius of the bangle be ’r’

Given, Circumference – Diameter=5 cm

Circumference of a circle = 2πr

Diameter of a circle = 2r

Therefore, \(2\pi r\) – 2r =5 cm

2r (\(\pi\)-1) = 5 cm

2r (22/7−1)

=5cm²r×15/7

=5r

=5×7/15×2 r

=1.166cm

Ques. A girl wants to create a square-shaped figure from a circular wire of a radius of 49 cm. Find the sides of a square. (3 Marks)

Ans. Let us take the radius of the circle to be ’r’.

Length of the wire=circumference of the circle=\(2\pi r\)

= 2×22/7×49

=2×22×7

=308cm

Again, let the side of the square be‘s’.

The perimeter of the square is equal to the length of the wire which is 4s

s=308/4

=77cm

Ques. Calculate what will be the area of a circular region whose radius is 21 m. (2 Marks)

Ans. The radius of the circular region, r = 21 m

Area of a circle = \(\pi r^2\)

= (22/7) × 21 × 21

= 22 × 3 × 21

= 1386 sq. m

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Radius Formula: Meaning, Calculation in Different Cases & Solved Example

What is Radius?

Radius Formula Detailed Video Explanation:

Finding the Radius of a Circle

Radius Formulas

Things to Remember

Solved Questions

CBSE CLASS XII Related Questions

1.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

2.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

3.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

5.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Similar Mathematics Concepts

Comments

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Radius Formula: Meaning, Calculation in Different Cases & Solved Example

What is Radius?

Radius Formula Detailed Video Explanation:

Finding the Radius of a Circle

Radius Formulas

Things to Remember

Solved Questions

CBSE CLASS XII Related Questions

1.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

2.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

3.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

4. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

5. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

2.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

3.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

5.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]