Home / United States / Math Classes / Properties of Circle

A circle is an important two dimensional shape in geometry. A circle is a closed curved figure formed by joining all the points that are equidistant from a fixed point. Here we will learn about the properties of a circle....Read MoreRead Less

A **circle** is the locus of a point which moves in such a way that it is at a fixed distance from a fixed point. The fixed point is called the **center** of the circle and the fixed distance is called the** radius** of the circle. Objects we see in everyday life like the wheels of a bicycle, a coin, a button, clocks and so on are in the shape of a circle.

The boundary of a circle is known as its ** circumference**.

**Radius (r):**The distance of the center from any point on the circumference is called the radius of the circle.**Diameter (d):**Diameter is a line segment drawn to join any two points on the circumference of a circle passing through its center. The diameter(d) is twice the measurement of the radius(r).

Hence, this shows us that, d = 2r.

**Chord:**Chord is a line segment made by joining any two points on the circumference of a circle.

*Note: The longest chord of a circle is called the diameter.*

**Secant:**A secant is a line that intersects the circle at two points.

**Tangent:**A tangent is a line that touches the circle at one point only. This point is called the point of contact.

**Arc:**An arc is a part of the circumference of a circle. The larger part of the circumference is called theand the smaller part is called the*major arc*.*minor arc*

- The diameter of a circle is always twice the radius.
- The longest chord of a circle is called the diameter.
- Circles are said to be congruent if they have the same radius.
- All circles are similar to each other.
- A tangent is always perpendicular to the radius at the point of contact with the circle.
- Circumference(C)
**:**The measure of total distance around a circle is called the circumference. The circumference of a circle is equal to the product of the constant π and the diameter of the circle.

Circumference of circle, C = \(\pi\)d

Also, circumference of circle, C = 2\(\pi\)r [d = 2r]

*[Note: The ratio of the circumference and diameter of a circle is always constant and is known as “**pi” **and denoted by the Greek letter**. *

*Hence, \(\frac{\text{circumference}}{\text{diameter}}=\pi\)**]*

- Area of circle(A): The region enclosed by the circumference of circle is called the
of a circle and is calculated by multiplying with the square of the radius (r).*area*

Area of circle, A = \(\pi r^2\)

In terms of the diameter, area of circle, A = \(\frac{\pi d^2}{4}\)

- Diameter is twice the radius (d = 2r).
- Circumference of circle, C = 2\(\pi\)r
- Circumference of circle, C = \(\pi\)d
- Area of circle, A = \(\pi r^2\)
- Area of circle, A = \(\frac{\pi d^2}{4}\)

**Example 1:** Find the diameter of a circle if its radius is 5 inches.

**Solution:** As we know, that the diameter is twice the radius, so,

d = 2r Write the formula

d = 2\(\times\)5 Substitute 5 for r

d = 10 Simplify

So, the diameter is 10 inches.

**Example 2:** Find the radius of a circle if its diameter is 22 cm.

**Solution:** As we know, that the diameter is twice the radius, hence,

d = 2r Write the formula

22 = 2\(\times\)r Substitute 22 for d

\(\frac{22}{2}=\frac{2 \times \text{r}}{2}\) Divide both side by 2

11 = r

So, the radius of the circle is 11 cm.

**Example 3:** Calculate the circumference of a circle if radius is 7 meters. (Use \(\frac{22}{7}\) for \(\pi\))

**Solution:**

C = 2\(\pi\)r Write the formula for circumference

C = 2\(\times \frac{22}{7} \times 7\) Substitute \(\frac{22}{7}\) for \(\pi\) and 7 for r

C = 44 Simplify

So, the circumference of the circle is 44 meters.

**Example 4: **Find the area of the circle if the radius is 21 feet in terms of \(\pi\).

**Solution: **

A = \(\pi r^2\) Write the formula for area of circle

A = \(\pi \times {21}^2\) Substitute 21 for r

A = 441\(\pi\) Simplify

So, the area of circle is 441\(\pi\) square feet.

**Example 5:** Calculate the area of a circular park if its diameter is 100 meters. (Use 3.14 for \(\pi\))

**Solution: **

A = \(\frac{\pi d^2}{4}\) Write the formula for area of circle

A = \(\frac{3.14 \times 100^2}{4}\) Substitute 3.14 for and 100 for d

A = 7850 Simplify

So, the area of the garden is 7850 square meters.

**Example 6:** Tom goes for a walk around the pond in a park. The pond is in the shape of a circle and has a radius of 1 mile. If Tom walks twice around the pond, then, calculate the distance he walks in total?

**Solution:**

The distance walked by Tom in one round is equal to the circumference of the circular pond.

C = 2\(\pi\)r Write the formula for circumference

C = \(2\times \frac{22}{7} \times 1\) Substitute \(\frac{22}{7}\) for \(\pi\) and 1 for r

C = \(\frac{44}{7}\) Simplify

So the distance walked by Tom in one round is \(\frac{44}{7}\) miles.

Total distance walked by Tom = \(2 ~\times\) distance walked in 1 round

= \(2 \times \frac{44}{7}\)

= 12.57

Hence, the total distance Tom walked is 12.57 miles.

Frequently Asked Questions

A circle can only be drawn on a plane so it is a two dimensional shape.

The ratio of circumference and diameter is a constant value called “pi” and is denoted by the Greek letter \(\pi\).

The value of pi is 3.141592653589793238……………, which is also non terminating and non recurring. So it is an ** irrational number**.

Half of a circle is called a semicircle.

Yes, with the use of circumference we can find out the measure of the radius, and then, by using the formula A = \(\pi r^2\), we can calculate the area of a circle.