Right from that peanut butter chocolate chip cookie that you're dreaming of (or will be dreaming of) right up to the pebble you see at the beach that you've been longing to visit. All of them have one thing in common.

What’s that?

They are circular in shape. Round things are an integral part of life. You can see many circular things around you.

Check out the interactive simulations to know more about circles and area of circle, and try your hand at solving a few interactive questions at the end of the page.

This mini lesson will help us explore the world of circles. The journey will take us through the definition of circle, to area of a circle formula, how to find the area of a circle, area of circle formula, area of a circle diameter, area of a circle equation, the surface area of a circle, area of a circle example, and area of circle circumference.

**Lesson Plan**

**What is Area of a Circle?**

The area of a circle is the amount of space enclosed within the boundary of a circle.

In the figures below, the colored region within the boundary of the circle is the area occupied by the circle.

**Area of a Circle Formula**

- Area of circle formula in terms of the radius is:

\( \text{Area of a Circle} = \pi \text{r}^2\) |

Area of the circle formula in terms of the diameter is:

\(\begin{align} \text{Area of a Circle} = \!\frac{\pi}{4} \!\times\! \text{d}^2\end{align}\) |

\(\text{r}\) is the radius of the circle and \(\text{d}\) is the diameter of the circle

\(\pi \) is the constant whose value is \( \approx \!\!\dfrac{22}{7} \!\! \: \text{or} \:3.14 \)

**Area of a Circle Equation**

Here's a simple simulation for you to interact with, along with explanations.

- As we can see in the simulation above, the circle can be cut into a triangle with the radius being the height of the triangle and the perimeter as its base which is

\[2 \pi r\] - We know that the area of the triangle is found by multiplying its base by the height and then dividing by 2, which is

\[\begin{align}\left(\frac{1}{2}\right) \times {2} \pi r \times r\end{align}\]

Therefore, the area of a circle is \(\pi\) times radius squared.

\(\text{Area of a Circle}\) \(= \pi r^2\) |

**Area of a Circle Using Diameter**

- Area of the circle in terms of the diameter is:

\(\begin{align} \text{Area of a Circle} = \!\frac{\pi}{4} \!\times\! \text{d}^2\end{align}\) |

**Area of a Circle Using Circumference**

We know that Circumference of a circle is given by the formula

\(\text{Circumference of a Circle}\) \(= 2 \pi r\) |

where r is the radius of the circle.

The area of a circle in terms of the circumference is given by the formula

\(\dfrac{\text{circumference }^2}{(4\pi)}\) |

**How to Calculate the Area of a Circle**

If we know the radius, the diameter, or the circumference of a circle, we can apply any of the formulas listed below and calculate the area of a circle.

Area of a Circle when the radius \(\left(r\right)\) is known | \(\pi r^2\) |
---|---|

Area of a Circle when the diameter \(\left(d\right)\) is known | \(\begin{align} \frac{\pi}{4} \!\times\! \text{d}^2\end{align}\) |

Area of a Circle when the circumferemce \(\left(\text{C}\right)\) is known | \(\dfrac{\text{C}^2}{4\pi}\) |

**Area of a Circle - Solved Examples **

Example 1 |

Help Tim how to find the area of a circle and its circumference whose radius is 14 in.

**Solution**

**Given:**** **Radius of the circle = 14 in

**Circumference of the Circle**

\[\begin{align} &= 2\pi r \\ & = 2 \times \left(\frac{22}{7}\right) \times \left({14}\right) \\ & = \text{88} \text{ in} \end{align} \]

\(\therefore\) Circumference of the Circle \(= 88 \text { in}\) |

**Area of a Circle **

\[\begin{align} &= \pi r^2 \\ &= \left(\frac{22}{7}\right) \times \left({14}\right) \times \left({14}\right) \\ &= \text{616} \text{ in}^2 \end{align} \]

\(\therefore\) Area of the Circle \(= 616 \text { in}^2\) |

Example 2 |

The ratio of the area of 2 circles is 4:9 Find the ratio of their radii.

**Solution**

Let us assume the following.

- Radius of the 1st circle = \(r_1\)
- Area of the 1st circle = \(A_1\)
- Radius of the 2nd circle = \(r_2\)
- Area of the 2nd circle = \(A_2\)

**Given:**** **\(A_1\) **:** \(A_2\) = 4 **:** 9

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

\[\begin{align} &⇒ \pi (r_1)^2 : \pi (r_2)^2 = 4:9 \\ &⇒ (r_1)^2 : (r_2)^2 =4:9 \\ &⇒ (r_1 : r_2)^2 = 4:9 \\ &⇒ (r_1 : r_2) = 2:3 \end{align} \]

\(\therefore\) Ratio of the Radii \(= 2:3 \) |

Example 3 |

A race track is in the form of a circular ring. The inner radius of the track is 58 yd and the outer radius is 63 yd. Find the area of the race track.

**Solution**

**Given:**

\(R\) = 63 yd

\(r\) = 56 yd

**Let the area of outer circle be **\(A_1\) and **the area of inner circle be **\(A_2\)

**Area of race track**

\[\begin{align} &= (\text A_1) - (\text A_2) \\ & = \pi(R)^2 - \pi (r)^2 \\ & = \pi (63^2 -56^2) \\ & = \left(\frac{22}{7}\right) \times 833 \\ & = 2618 \text { yd}^2 \end{align} \]

\(\therefore\) Area of the Race Track \(= 2618 \text { yd}^2\) |

Example 4 |

A wire is in the shape of an equilateral triangle. Each side of the triangle measures 7 in. The wire is bent into the shape of a circle. Find the area of the circle that is formed.

**Solution**

**Perimeter of the Equilateral Triangle**

\[\begin{align} &= \text{3} \times \text{Side} \\ & = 3 \times 7 \\ & = 21 \text { in} \end{align} \]

\(\therefore\) Perimeter of the Triangle \(= 21\text { in}\) |

Since perimeter of the equilateral triangle = Circumference of the circle formed,

**Circumference of a Circle \( =2 \pi r\) **

\[\begin{align} 2 \pi r &= 21 \text { in} \\ r&=3.34 \text { in} \end{align} \]

Let us keep the above result in that form.

Therefore, **Radius of the circle is 3.34 cm.**

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

\[\begin{align} &= \left(\frac{22}{7}\right) \times (3.34)^2 \\ &=35.06\text{ cm}^2 \end{align} \]

\(\therefore\) Area of the Circle \(= 35.06 \text { in}^2\) |

Example 5 |

The time shown in a circular clock is 3:00 pm. The length of the minute hand is 21 in. Find the distance travelled by the tip of the minute hand when the time is 3:30 pm.

**Solution**

When the minute hand is at 3:30 pm, it covers half of the circle.

So, the distance travelled by the minute hand is actually half of the circumference.

Distance \(= \pi r\) (where \(r\) is the length of the minute hand)

\[\begin{align} &= \left(\frac{22}{7}\right) \times (21) \\ & =66\text{ in} \end{align} \]

\(\therefore\) Distance Travelled \(= 66 \text { in}\) |

- How would you find the area of a sector of a circle?
- To warn ships of underwater rocks, a lighthouse spreads a red-coloured light over a sector of angle 80°. The light reaches a distance of 16.5 km. Find the area of sea over which the ships are warned.
- Tom is riding a cycle to his friend's house. The wheel of his cycle has a radius of 20 in, and his friend lives 660 yd away. How many times does his front wheel rotate when he goes and comes back from his friend's house?

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

1. | The radii of the circles are given as 1, 2, 3, 4, and 5 in. Help Tim Find the area of the shaded region. |

**Let's Summarize**

We hope you enjoyed learning about **area of a circle** with the simulations and practice questions. Now you will be able to easily solve problems on area of a circle formula, how to find the area of a circle, area of circle formula, area of a circle diameter, area of a circle equation, the surface area of a circle, area of a circle example, and area of circle circumference.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions(FAQs)**

## 1. How do you find the area of a circle?

The area of a circle is \(\pi\) multiplied by the square of the radius. It can be found using the formula given below:

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

Where \(r\) is the radius of the circle and \(\pi \approx 3.14\)

## 2. How do you find the circumference and area of a circle?

The area and circumference of a circle can be calculated using the formulas below:

Circumference = \(2 \pi r\)

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

## 3. How to calculate the area of a circle with the diameter?

Use the formula \(\begin{align} \text{Area of a circle} = \!\frac{\pi}{4} \!\times\! \text{diameter}^2\end{align}\)

## 4. What is the area of a circle with a diameter of 6 units?

We know that \(\begin{align} \text{Area of a Circle} = \!\frac{\pi}{4} \!\times\! \text{Diameter}^2\end{align}\)

Substituting diameter = 6 we get

\(\begin{align} \text{Area of a Circle} \\

&= \frac{\pi}{4} \!\times\! \text{6}^2 \\

&=\frac{\pi}{4} \times 36 \\

&=9 \:\pi \\

&= 9 \times \frac{22}{7} \\

&\approx28.274\text{sq. units}

\end{align}\)