A circle is a two-dimensional shape where all points on the circumference are the same distance from the centre.
- It consists of all points in a plane that are equidistant (at the same distance) from a fixed point called the centre.
- The distance from the centre to any point on the circle is called the radius.
- Its area is equal to pi times the square of its radius, or πr2.
Various objects that we observe in real life are circular in shape. Some examples of circular-shaped objects are Chapattis, Coins, Wheels, Rings, Buttons, CDs/DVDs, Bangles, Plates, etc.
Interior and Exterior of Circle
If we draw a circle, it divides the 2-D plane into three parts, which are:
| Position | Description |
|---|---|
| Inside the Circle | A point whose distance from the centre is less than the radius is called a point inside a circle. |
| On the Circle | Points whose distance from the circle's centre is equal to the radius; these lie on the circumference. |
| Outside the Circle | Points whose distance from the circle's centre is greater than the radius are known as exterior points. |
Properties of a Circle
Some of the properties of the circle are
- Circles with the same radius are called congruent circles.
- Equal chords are equidistant from the centre of the circle.
- Equidistant chords from the centre of the circle are always equal.
- The perpendicular drawn from the centre of the circle to the chord always bisects the chord.
- We can draw two tangents from an external point to a circle.
- Tangents drawn from the endpoints of the diameter are always parallel to each other.
Circle Formulas
There are various formulas related to the circle. Let the radius of the circle be 'r'; then some of the important formulas related to the circle are
| Formulas of Circle | Expressions |
|---|---|
| Area of Circle | πr2 |
| Circumference of Circle | 2πr |
| Length of Arc of Circle | θ × r |
| Area of Sector of Circle | (θ × r2) / 2 |
| Length of Chord | 2r sin(θ/ 2) |
| Area of Segment | r2(θ - sinθ)/2 |
Note: Value of π is taken to be 3.14 or 22/7
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Solved Examples
Example 1: If the diameter of a circle is 142.8 mm, then what is its radius?
Diameter = 142.8 mm
By Formula,
Diameter = 2 radiusRadius = (142.8 ÷ 2) = 71.4 mm
Thus, radius of circle is 71.4 mm
Example 2: Distance around a park is 21.98 y d. What is the radius of the park?
Circumference of the Park = 21.98 yd
We know that,
Circumference = 2π × Radius
Radius = Circumference / 2πRadius = 21.98 / 2×3.14 = 3.5
Thus, radius of circle is 3.5 yd
Example 3: The inner levelling circumference of a circular track is 440 m, and the track is 14 m wide. Calculate the cost of levelling the track at 25 rupees/m2.
Given:
Inner circumference = 440 m, Track width = 14 m, Rate = Rs. 25 per m²Step 1: Find inner radius
2 × (22/7) × r = 440
r = (440 × 7) / (2 × 22)
r = 3080 / 44
r = 70 mStep 2: Find outer radius
R = 70 + 14 = 84 mStep 3: Find area of track
Area = (22/7) × (R² − r²)
Area = (22/7) × (84² − 70²)
Area = (22/7) × (7056 − 4900)
Area = (22/7) × 2156
Area = 6776 m²Step 4: Find cost
Cost = Area × Rate
Cost = 6776 × 25
Cost = Rs. 1,69,400
Example 4: Find the length of the chord of a circle where the radius is 8 cm, and the perpendicular distance from the chord to the centre is 3 cm.
Given,
Radius, r = 8 cm
Distance of Chord to Centre, d = 3 cmChord Length = 2√(r2 - d2)
= 2√(82 - 32)
= 2√(64 - 9)
= 2√55
= 2 × 7.416Chord length = 14.83 cm
Thus, length of chord is 14.83 cm