Have you ever noticed the shape of a pizza?
What is this shape called?
You are right!
It is of circular shape.
Circles are one of the most commonly found shapes in the world. The fascinating properties of circles make it an important topic in geometry.
In this chapter, you will learn about circle diameter, area of a circle, circle calculator, circumference, and area of circles, in the concept of Circle. Check out the interactive simulation on the formation of a circle.
Try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
What is Circle?
A circle is formed by the set of points which are at a constant or fixed distance (radius) from a fixed point (center) in the plane.
The fixed point is called the origin or center of the circle.
The fixed distance of the points from the origin is called the radius.
Shape of a Circle
Let's take a fixed point \(P\) and try to draw lines from it.
Let's draw several lines of length 3 cm away from point \(P\) in all possible directions.
When we join these ends together in a curve, it forms a circle.
Try this simulation which shows how the shape of a circle is formed by the collection of points which are equidistant from the center.
(Click on the buttons: Reset and See)
The circle we have drawn resembles the wheel of a cart.
Have you ever ridden a ferris wheel at a carnival?
Did you find anything common in these items?
Yes, it's their shape, which is a circle.
Properties of a Circle
The fascinating properties of circles makes it an important topic in math and geometry.
Imagine a circular park in your neighborhood.
Try to identify the various parts of a circle with the help of the figure and table given below.
| In a Circle | In our park | Named by the letter |
|---|---|---|
| Centre | Fountain | \(F\) |
| Circumference | Boundary | |
| Chord | Play area entrance | \(PQ\) |
| Radius | Distance from the fountain to the Entrance gate | \(FA\) |
| Diameter | Straight Line Distance between Entrance Gate and Exit Gate through the fountain | \(AFB\) |
| Minor segment | The smaller area of the park, which is shown as the Play area | |
| Major segment | The bigger area of the park, other than the Play area | |
| Interior part of the circle | The green area of the whole park | |
| Exterior part of the circle | The area outside the boundary of the park | |
| Arc | Any curved part on the circumference. |
Could you find these parts in the park?
Well done!
Parts of a Circle
Radius
Radius is the distance from the center of a circle to the boundary of the circle.
Let's refer to the park that was shown earlier.
A circle can have multiple radii as they all start from the center and touch the boundary of the circle at various points.
Diameter
Diameter is a straight line passing through the center that connects two points on the boundary of the circle.
Let's go back to the figure of the park.
We can see that \(AFB\) is the diameter.
We should note that there can be multiple diameters in that park, but they should:
- pass through the fountain (center) in a straight line
- touch one boundary of the park to the other
Chord
Any line segment that touches the circle at two different points is known as the chord of a circle.
Referring to the park, we can see that \(PQ\) is the chord that shows where the play area starts.
Circumference
The circumference of a circle is its boundary or the length of the complete arc of a circle.
Considering the same park, if you jog around the boundary of that park and complete one full circle without going anywhere else, that complete round is the circumference of a circle.
- The diameter is the longest chord in a circle.
-
If the radius of a circle is extended further and touches the boundary of the circle, it becomes the diameter of a circle. Therefore, Diameter \(= 2 \times \) Radius.
-
The circle's circumference is the distance around a circle or length of a circle. We can find the circumference of a circle using the radius or diameter.
-
Geometric shape of a circle has zero sides.
Circle Formulas
Here are the three basic circle formulas that are used in calculating the various dimensions of a circle.
If the radius is known, the diameter can be calculated by multiplying it by 2
| Diameter of a circle \(= 2 \times r \) |
If the diameter is known, the radius can be calculated by dividing it by 2
| Radius of a circle \(= \frac{\text{d}}{2}\) |
The circumference of a circle can be calculated using the radius of the circle and the value of pi.
| Circumference of a circle \(= 2\pi r \) |
The area of a circle is the region occupied by the circle on a two-dimensional plane, which can be calculated using the radius and the value of pi.
| Area of a circle \( = \pi {r^2} \) |
\(r\) is the radius of the circle and \(d\) is the diameter of the circle.
The value of\(\begin{align}\pi = {\frac{22}{7}} \end{align}\)
Circle Calculator to Calculate Area of Circle
Use the circle calculator to observe the behavior of the area with other parameters of a circle.
Solved Examples
Let us see how the shape of a circle is seen as a part of our everyday lives.
| Example 1 |
David was playing near a swimming pool.
He threw his ball which splashed in the water and floated away.
With reference to the exterior and interior of a circle, can you spot his location and the place where his ball is floating?
Solution
| Subject | Position |
|---|---|
| David | Exterior part of the circle |
| His ball | Interior part of the circle |
This one was simple!
Were you able to find his location?
Good job!
| Example 2 |
Identify the parts of a circle hidden in this wheel.
Find the radius, the diameter, and the circumference with the help of the given colors and clues.
Solution
| Parts hidden in the wheel | Parts of a Circle |
|---|---|
| \(OC\) (red) | Radius |
| \(HD\) (green) | Diameter |
| The black boundary | Circumference |
| Example 3 |
Do you like cake?
Most people do.
Peter bought a circular chocolate cake from his favorite bakery.
He cut it into two parts, but the portions could not be cut equally.
Can you identify the major segment, the minor segment, the chord, and the arc in it?
Solution
The smaller part of the cake is the minor segment.
The bigger part of the cake is the major segment.
The curve is the arc.
The line which separates the cake into two parts is the chord.
| Example 4 |
John went swimming in a circular swimming pool.
After swimming, he runs one round along the boundary of the pool.
If the radius of the pool is 35 feet, can you find the distance that John ran around the pool?
To find the distance that John ran, you need to know the circumference of the circle (pool).
For this, you need to know the value of \(\pi\) and \(r\), where ‘\(r\)’ is the radius of the pool.
Given:
- \( r = 35 \) feet
- \(\begin{align}\pi = {\frac{22}{7}} \end{align}\)
Using the formula, Circumference of Circle (C) \(= 2\pi r\)
\[\begin{align}C &= 2 \times {\frac{22}{7}} \times 35 \\\\ &= 220 \text{ feet}\end{align}\]
| \(\therefore\) John ran \(220\) feet |
| Example 5 |
You want to decorate your tabletop, which is in the shape of a circle, with a colorful sticker.
If the radius of the tabletop is 21 in, find the amount of paper you need to cover its top surface.
Solution
Given,
- Radius of the table top \(=\) 21 in
- \(\begin{align}\pi = {\frac{22}{7}} \end{align}\)
Using the formula, Area of Circle (A) \(= \pi r^2\)
\[\begin{align}A &= {\frac{22}{7}} \times 21 \times 21 \\\\ &= 1386 \text{ sq.in} \end{align}\]
| \(\therefore\) Area of table top \(= 1386 \text{ sq.in} \) |
- The value of \(\pi \) can be approximated as \(\begin{align}22\over 7 \end{align} \) when the radius of the circle is divisible by 7. Otherwise, it can be approximated to 3.14. This can be used to calculate the circumference and the area of the circle.
- While calculating the area of a circle it is helpful to write the radius twice instead of squaring it. Squaring will make the calculations tedious. This holds true for radius with larger values. For example, if the radius of a circle is 21 in, it will be easier to calculate \(\begin{align}{22\over 7} \times 21 \times 21 \end{align} \) than calculating \(\begin{align}{22\over 7} \times 441 \end{align} \)
Interactive Questions
Here are a few problems related to Circle.
Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about Circle with the simulations and practice questions. Now you will be able to easily solve problems on circle diameter, area of a circle, circle calculator, circumference, and area of circles.
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Frequently Asked Questions (FAQs)
1. What is the circumference of a circle with diameter 6 inch?
Circumference of Circle (C) \(\begin{align} &= \pi \times D \\[0.2cm]&= \frac{{22}}{7} \times 6 \\[0.2cm]&= \frac{{132}}{7}inch \\\end{align}\)
2. What tool do you use to measure the diameter?
A Caliper is a tool used to measure the inside and outside distance (diameter) of an object.
3. What is the circumference of a circle when the radius is 4 feet?
Circumference of Circle (C) \(\begin{align}& = 2\pi r \\[0.2cm] &= 2 \times {\frac{22}{7}} \times 4 \\\\ &= 8 \times {\frac{22}{7}} \\\\ &= \frac{{176}}{7}\text{ feet}\end{align}\)