Radius

Introduction to Circles 

A circle is a collection of points on a two-dimensional plane, which are equidistant from the centre point '\(O\)'.

 a collection of points equidistant from the center

The distance from the center point to any endpoint on the circle is the radius.

The size of the circle changes when the length of the radius varies.

You can observe this change by dragging the blue endpoint outwards on the circle.

 


What is a Radius?

The radius of a circle definition is the length of the line segment from the center of a circle to a point on the circumference of the circle. 

A circle can have many radii (the plural form of radius) and they measure the same.

Radius of a circle is generally abbreviated as ‘\(r\)’. 

Look at this image:

Circle marked with radii

The points \(A,\,B,\,M,\,N,\,P,\,Q,\,X,\,Y\) lie on the boundary of the circle.

These points are equidistant from the center \(O\).

\(OA=OB=OM=ON=OP=OQ=OX=OY\)

Diameter of a Circle

Diameter is the longest chord of a circle.

The radius of a circle definition can be used to understand the diameter of a circle.

The diameter is twice the length of the radius.

Diameter of a circle\(=2r\)

Circumference of a Circle

The perimeter of the circle is called the circumference. It is the boundary of the circle. 

Look at this simulation to understand the relationship between the radius and the circumference of a circle.

Drag the radius between \( 1 \: \text{cm}\) to \(5 \:\text{cm} \).

 

The circumference of a circle in terms of its radius is expressed as: 

\(C = 2\pi r\)

Where;

  • \(C\) is the circumference
  • \(r\) is the radius
  • \(\pi\) is the constant \(3.14159...\)

Area of a Circle

Area of a Circle is the area occupied by the circle.

The relationship between the radius and area is given by:

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

Where;

  • \(r\) is the radius
  • \(\pi\) is the constant \(3.14159...\)

Formulas for Radius

Radius can be calculated when the diameter, the area, or the circumference is known. Refer to the formulas below to calculate the radius.

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

\(\begin{align}\text{Radius}= \frac{\text{Diameter}}{2} \end{align}\)

The radius of a circle formula when the circumference is known is:

  \(\begin{align}\text{Radius}= \frac{\text{Circumference}}{2\pi}\end{align} \)

The radius of a circle formula when the area is known is: 

\(\begin{align}\text{Radius}= \sqrt \frac{\text{Area of the circle}}{\pi} \end{align}\)

Use the radius of a circle calculator in the next section to observe the behaviour of radius with other parameters of a circle.


Radius of a Circle Calculator

Use this calculator to understand the properties of a circle.

Input the radius, click Go and see how other parameters like diameter, circumference, and area are related to it.

The radius of a circle calculator will help you understand the relations between them. 


Equation of a Circle 

Look at the diagram of a circle in the cartesian plane. The coordinates of the center are \( (h,k) \).

Equation of a circle shown on a cartesian plan with coordinates

The radius of a circle equation in the cartesian coordinate plane is given by:

\((x−h)^2 + (y−k)^2 = r^2\)

\((x,y)\) are the points on the circumference of the circle which are at a distance ‘\(r\)’ (radius) from the center \( (h,k) \).

When the center of the circle is at origin \((0,0)\), the equation of the circle reduces to:

\[x^2 + y^2 = r^2\]


Radius of a curve

Arc of the circle is the distance between two points along a section of a curve. The radius of the curve is the radius of the circle of which it is a part.

Radius of a curve

The formula to find the radius when the length of the chord defining the base (W) and the height (H) measured at the midpoint of the arc's base is given is:
  

\[\text{Radius}= \frac{\text{H}}{2} +\frac{\text{W}^2}{8\text{H}} \]

 

 
important notes to remember
Important Notes
  1. \(\begin{align}\text{Radius}= \frac{\text{Diameter}}{2} \end{align}\)
  2. \(\begin{align}\text{Radius}= \frac{\text{Circumference }}{2\pi} \end{align} \)
  3. \(\begin{align}\text{Radius}= \sqrt \frac{\text{Area of the circle}}{\pi} \end{align} \)
  4. \(\begin{align}(x−h)^2 + (y−k)^2 = r^2 \end{align} \)
  5. \(\begin{align}\text{Radius}= \frac{\text{H}}{2} +\frac{\text{W}^2}{8\text{H}}  \end{align} \)

Solved Examples

Example 1

 

 

The radius of the given circle is \(5 \text{ cm}\). Find its circumference and area. (Use \(\pi \!= \!3.14\))

 Circle of radius 5 cm

Solution:

We know that radius of the given circle \(=5\ \text {cm}\)

We know that circumference 

\[\begin{align}
&= 2\pi r \\
&= 2\times \pi \times 5 \\
&=2\times 3.14 \times 5 \\
&=31.4\:\text{cm}
\end{align}\]

We know that area of the circle

\[\begin{align}
&= \pi r^2\\
&=(\pi \times 5 \times 5)\\
&=25 \times 3.14 \\
&=78.5\: \text{cm}^2
\end{align}\]

\( \therefore\)  Circumference \(=31.4\:\text{cm} \) 

and

Area of the circle \(=78.5\: \text{cm}^2\)

Example 2

 

 

 The area of a circle is \(12.56\: \text{cm}^2\). What is its diameter? (Use \(\pi = 3.14\))

Circle of area 12.56 square centimeter

Solution:

We know that the area of the circle 

\[\begin{align}
&= \pi r^2\\
\therefore 12.56&=(\pi \times r^2)\\
12.56&=\text{r}^2\times 3.14 \\
r^2&=\frac{12.56}{3.14} \\
r&=\sqrt{\frac{12.56}{3.14}}\\
r&=\sqrt{4}\\
r&=2 \:\text{cm}
\end{align}\]

We know that diameter of a circle

\[\begin{align}&= 2r\\
&= 2 \times 2 \\
&=4 \:\text{cm}
\end{align}\]

\( \therefore\) Diameter of the circle \(= 4 \text{ cm}\)
Example 3

 

 

Amy broke a circular disc exactly into two halves. The radius of the disc is \(14 \:\text{cm}\). What is the perimeter of the broken half-disc? Use \(\left(\pi = \frac{22}{7}\right)\)

 a circular disc of radius 14cm

Solution:

We know that radius of the given disc

\[=14\ \text {cm}\]

We know that circumference 

\[\begin{align}
&= 2\pi r \\
&= 2\times \pi \times 14 \\
&=2\times \frac{22}{7}\times 14 \\
&=88\:\text{cm}
\end{align}\]

\(88\: \text{cm}\) is the circumference of the full disc .

The perimeter of the broken disc which is semi-circular in shape is the sum of half the perimeter of the circle and the diameter \[(\pi r\! +\!\text{AB})\].

perimeter of semi circular disc

\[\begin{align}
&= \frac{88}{2} +14+14\\
&= 28+44 \\
&=72 \\
&=58\:\text{cm}
\end{align}\]

\( \therefore\) Perimeter of broken half-disc \(=72\:\text{cm} \)
Example 4

 

 

Find the radius of the circle whose center is O\((2,1)\), and the point P\((5,5)\) lies on the circumference.

  a circle in Cartesian coordinate

Solution:

The equation of a circle in the cartesian plane is given by: 

\[(x−h)^2 + (y−k)^2 = r^2\]

Substituting the value of \((x,y)\)  as \((5, 5)\) and \((h,k)\) as \((2,1)\) we get 

\[\begin{align}
&= (5−2)^2 + (5-1)^2 = r^2\\
&= (3)^2 + (4)^2= r^2\\
&= 9+16 =r^2\\
r^2&= 25\\
\therefore r&=5
\end{align}\]

\( \therefore\) Radius of the given circle is \(5 \:\text{units}\)
Example 5

 

 

The dimensions of the segment of a circle are as shown in the figure. Find the radius of the curve.  

radius of a curve

Solution:

Radius of the curve

\[ = \frac{\text{H}}{2} +\frac{\text{W}^2}{8\text{H}} \]

Given 

\[\begin{align}
\text{W} &= 8\\
\text{H}&=4
\end{align}\]

Substituting the above values in the radius of a curve formula, we get:

\[\begin{align}
 &= \frac{ {4}}{2} +\frac{8^2}{8\times\text{4}}  \\
 &=2 +\frac{64}{32}\\
&=2+2\\
&=4 \end{align}\]

\( \therefore\) Radius of the curve is \(4\: \text{cm}\)

Practice Questions

 
 
 
 
 
 
 
Challenge your math skills
Challenging Questions
  1.  Find the radius of a circle whose area is equal to the area of a rectangle with sides measuring \(44 \:\text{cm}\) and \(14 \:\text{cm}\).
  2. The ratio of radii of two circles is \(2:3\). Find the ratio of their circumference. 
  3. John cut out a circle of radius \(5 \:\text{cm}\) from a square sheet of side \(10 \:\text{cm}\). What is the area of the remaining sheet?

Maths Olympiad Sample Papers

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.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. How do you find the radius of a circle?

The radius of a circle can be found when the diameter, circumference, or the area are known.

  • \(\begin{align}\text{Radius}= \frac{\text{Diameter}}{2} \end{align} \)
  • \(\begin{align}\text{Radius}= \frac{\text{Circumference }}{2\pi} \end{align} \)
  • \(\begin{align}\text{Radius}= \sqrt \frac{\text{Area of the circle}}{\pi} \end{align} \)

2. How to calculate the radius of a circle?

If the diameter of the circle is known, we can calculate the radius using the formula: \[\text{Radius}= \frac{\text{Diameter}}{2}\]

If the circumference is known, we can apply the formula: \[\text{Radius}= \frac{\text{Circumference }}{2\pi}\] and calculate the radius.

If the area of the circle is known, we can apply the formula: \[\text{Radius}= \sqrt \frac{\text{Area of the circle}}{\pi}\] and find the radius.

3. What is radius of a curve?

The radius of a curve is the radius of the circle of which it is a part of.

The formula to find the radius when the length of the chord defining the base (W) and the height measured at the midpoint of the arc's base (H) is: 

\[\text{Radius}= \frac{\text{H}}{2} +\frac{\text{W}^2}{8\text{H}} \]

  
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