About Properties of a Circle
What is a Circle?
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.
Important Terms used in Circle
- 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 the major arc and the smaller part is called the minor arc.
Properties of a Circle
- 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 area of a circle and is calculated by multiplying with the square of the radius (r).
Area of circle, A = \(\pi r^2\)
In terms of the diameter, area of circle, A = \(\frac{\pi d^2}{4}\)
Rapid Recall
- 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 = \(\frac{\pi d^2}{4}\)
Solved Examples
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.
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.