What are Circles?
The word circle comes from the Latin word ‘circulus’, which means a small ring. A circle is a two dimensional curved plane figure. All points on the boundary of a circle are equidistant from a fixed point within the circle. This fixed point is known as the center of the circle. If a circle is divided into two halves, each half is called a ‘semicircle’.
Parts of Circles
Radius of circle: The distance between the center to any point on the boundary of a circle is called its radius.
Diameter: Diameter is the distance between two points on the boundary of a circle through its center. The diameter of a circle is twice its radius.
Circumference: The circumference is the measure of length of its circular boundary.
Chord of a circle: A chord is a line segment that connects two different points on the circumference of a circle.
[Note: Diameter is the longest chord in a circle.]
Tangent: A tangent is a line that lies outside the circle and touches the circle at a single point.
Secant: The secant is a line that passes through two points on the circumference of a circle.
Arc of a circle: A curve that is a part or a portion of the circumference of a circle is referred to as an arc of the circle.
Segment in a circle: A segment in a circle is the area enclosed by a chord and the associated arc. There are two types of segments: minor segments and major segments.
Sector of a circle: A sector is the region enclosed by two radii and the associated arc. The two different sectors of a circle are the minor sector and major sector.
How to Calculate the Area and Circumference of a Circle?
Area of a circle is the amount of space enclosed within the circle. It can be calculated using the following formula:
Area of a Circle (A) = \(\pi r^2\)
Here, the term ‘r’ is the radius of the circle and \(\pi\) is a mathematical constant.
The circumference of a circle is defined as the total boundary length of the circle. The formula used to find the circumference of a circle is:
Circumference of Circle = 2\(\pi\)r = \(\pi\)d
[Note: Area is measured in square units and circumference is measured in the units of length.]
Properties of a Circle
Let’s look at the properties of a circle.
- A circle is a closed 2D shape that has one curved edge.
- If two circles have the same radius, we can say that the circles are congruent to each other.
- Equal chords are always at an equal distance from the center of the circle.
- The perpendicular bisector of the diameter passes through the center of the circle.
- The tangents that are drawn at the endpoints of the diameter will be parallel to each other.
Solved Examples
Example 1: A cow is tethered by a 5.5m long rope to a pole, which is at the center of the field. Find the maximum area that the cow can graze.
Solution:
It is stated that the length of the rope is 5.5m long.
Therefore, the maximum area that the cow can graze will be equal to the area of a circle whose center is the pole and radius is 5.5m.
We can find the total area by using the formula:
Area of Circle = \(\pi r^2\)
= 3.14 x (5.5)\(^2\) [Substitute 3.14 for \(\pi\) and 5.5 for r]
= 3.14 x (\(\frac{55}{10}\))\(^2\) [Write 5.5 as \(\frac{55}{10}\)]
= 3.14 x \(\frac{3025}{100}\) [Square \(\frac{55}{10}\)]
= \(\frac{9498~. ~5}{100}\) [Multiply]
= 94.985 [Simplify]
Therefore, the maximum area the cow can graze is 94.985 square meter.
Example 2: A copper wire of length 140 centimeters is bent into a circle. What will be the area of the circle formed?
Solution:
As stated, the length of the copper wire = 140 cm
Since the copper wire has been bent into a circle, the total length of the wire is the circumference of the circle formed. We can find the radius of the circle from its circumference.
That is,
Circumference of circle = 2\(\pi\)r
140 = 2 x 3.14 x r [Substitute the given values]
\(\frac{140}{6.28}\) = r [Solve for r]
\(\frac{140}{6.28}~\times~\) \(\frac{100}{100}\) = r [Simplify]
\(\frac{14000}{628}\) = r [Simplify further]
r = 22.3 [Divide]
So, the area of circle can be calculated using the formula:
Area of circle = \(\pi r^2\)
= 3.14 x (22.3)\(^2\) [Substitute 3.14 for \(\pi\) and 22.3 for r]
= 3.14 x 497.29 [Square]
= 1561.49 [Multiply]
Therefore, the area of circle formed by the wire is 1561.49 square centimeters.
Example 3: If the radius of a circle is 36 cm, what will be the length of its diameter?
Solution:
It is stated that radius (r) = 36 cm
As we know, the diameter of a circle is twice the radius.
Then, Diameter (d) = 2 × r
= 2 × 36 [Substitute 36 for r]
= 72 [Multiply]
Thus, the diameter of the given circle is 72cm.
Real-world examples of circles include camera lenses, pizzas, clocks, rings, steering wheels, plates, and so on.
Yes, a circle is a closed two-dimensional shape.
The parts of a circle include tangent, chord, radius, diameter, minor arc, major arc, minor segment, major segment, minor sector, and major sector.
In geometry, half of a circle is referred to as a semicircle.