What is a Circle?
The wheel’s invention paved the way for the development of the circular concept and its features. Many things we come across in our daily lives have a circular shape to them. Wheel, bangle, ring, cake, donuts, pie, button, clock, and pizza are a few examples. Regardless of the fact that all of the items are circular in shape, their geometrical measurements differ. Let’s have a look at the geometrical idea of a circle.
The word “circle” comes from the Greek word “kirkos,” which corresponds to “ring” or “hoop.” A circle is a 2-dimensional shape that has one curved edge and no corners. Circle is formed by joining infinite number of sides. A circle is defined as the set of points marked at an equal distance from the center. Center can be denoted using any alphabet, here it is marked as C, refer to the image below.
Parts of a Circle
The length of a curve around a circle is called circumference. A line segment joining any point on the circle from the center is called radius; it is denoted by ‘r’. A line segment joining two points on the circumference of a circle that divides circle into two equal halves is called diameter; it is denoted by ‘d’. Diameter is twice the radius. Refer to the image below.
Relationship between Radius, Diameter, and Circumference
Diameter, d = 2r = 2 times radius
Circumference, C = 2πr = πd (π = 3.14)
Example 1: Find the diameter of the circle whose radius is 5 cm.
Solution: d = 2r = 2 × 5 = 10 cm
Example 2: Find the radius of the circle whose diameter is 14 cm.
Solution: d = 2r ⇒ r = d/2
r = 14/2 = 7 cm
Example 3: Find the circumference of the circle whose radius is 10 cm. Use π = 3.14.
Solution: r = 10 cm, C = 2πr = 2 × 3.14 × 10 = 62.8 cm
Circumference, C = 62.8 cm.
Example 4: Find the radius of the circle whose circumference is 94.2 cm. Use π = 3.14.
Solution: C = 2πr = 94.2
⇒ r = C/2π ⇒ 94.2 /(3.14 × 2) = 15 cm
Therefore, radius r = 15 cm.
Chord of a circle
A chord of a circle is a straight line segment with both the endpoints lie on the circumference of the circle. Diameter is the biggest chord of the circle.
Angles in a circle
An angle in a circle is formed between two radii (plural of radius), chords, or tangents. There are four types of angles in circle. They are:
Central angle
An angle is formed at the intersection of two radii is called central angle. The vertex of the angle lies at the center of the circle.
Inscribed angle
An angle formed between two chords of the circle is called inscribed angle. The vertex of the angle lies on the circumference of the circle.
Interior angle
An angle formed between two lines at their intersection point inside the circle is called an interior angle.
Exterior angle
An angle formed between secants or tangents of the circle is called exterior angle. The vertex of the angle lies outside the circle.
Exterior angle of a circle
Arc of a circle
An arc of a circle is a portion of a circumference. Every circle has two arcs: Major arc and Minor arc.
Arc Notation
Arc length can be found using the formula given in the image below.
| Formula List to find Angle in a Circle | |
| (Arc length × 360)/2πr r is a radius of a circle. |
|
| ½ × intercepted arc | |
| ½ (a + b) a and b are intercepted arcs. |
|
| ½ (b – a) a and b are intercepted arcs. |
Let us learn to find the angles in a circle.
Example 5: Find the arc length of a circle whose inscribed angle is 0.79 radians and radius is 6 cm.
Solution: Arc length s = r × θ = 6 × 0.79 = 4.74 cm
Example 6: Find the arc length of a circle whose central angle is 84° and radius is 8 inches. Use π = 3.14.
Solution: Arc length s= θ × 180×r
s = 84 × (3.14 / 180) × 8 = 11.72 inches
Example 7: The length of the minute hand is 8.2 cm.
a) Find the area swept by a clock’s minute hand in one hour.
b) What is the area swept in 30 minutes and 3 hours by the same hand?
Solution:
a) The minute hand sweeps one full circle in one hour.
Radius of the circle = Length of the minute hand = 8.2 cm
Area swept in one hour = Area of full circle = πr2
Thus, the area swept in one hour = 3.14 × 8.22 = 211.13 cm2
The area swept in one hour is 211.13 cm2.
b) Area swept in 30 minutes = 211.13 ÷ 2 =105.57 cm2
Area swept by the same hand in 3 hours = 211.13 × 3 = 633.39 cm2
Thus, area swept by the minute hand in 30 minutes and 3 hours is 105.57 + 633.39 = 738.96 cm2.
Example 8: A circular mirror has a 4 cm frame around it. The mirror itself has a diameter 18 cm. Find the area of the frame.
Solution: Diameter of the mirror = 18 cm
Radius of the mirror = 18 ÷ 2 = 9 cm
Area of the mirror = πr2 = π × 92 = 81π cm2
Area of a mirror and frame:
Radius = 4 + 9 = 13 cm
Area = πr2 = π × 132 = 169 π cm2
Area of frame = 169 π – 81 π = 88π cm2 = 88 × 3.14 = 276.32 cm2
Sector of a Circle:
A portion of a circle enclosed by an intercepted arc and two radii is called sector. A pizza slice is one of the examples of a sector. A circle is divided into two sectors: Major sector and Minor sector.
Central angle decides whether the given sector is a major or a minor sector. A sector which has central angle less than 180° is called minor sector. A sector which has central angle more than 180° is called major sector.
Segment of a Circle
A region which is enclosed by the intercept arc and chord of a circle is called segment of the circle. There are two types of segments in a circle, major segment and minor segment.
Tangent of a Circle
A straight that touches the circle at only one point is called tangent of a circle. The point at which the tangent and circle meet is called the point of tangency.
When two or more circles intersect at one point is called tangent circle. Depending on the intersection of circles they are called internal tangents and external tangents.
Tangent circles can have same tangent called common tangent. Common tangents are drawn as shown in the image below. The tangent which passes through the space between two circles is called common internal tangent. The tangent which passes on the top or at the bottom of two circles is called common external common tangent.
Congruent and Concentric Circles
What do you observe when you cut onion horizontally? You get circular layers of onions. These layers can be arranged back one inside the other. These rings have a same center but different radius. Circles with the same center, but different radii are called concentric circles.
What do you observe in Olympic rings? You can see that all five rings have same dimensions. Circles with same radii, but different centers are called congruent circles.
Try it yourself!
Look around and identify the congruent and concentric circles. How many can you identify?
Semicircle
A shape obtained when a circle is cut along the diameter is called semicircle. Dome, protractor, and rainbow are of the best examples of semicircle.
Properties of a Circle
There are various features of circles in mathematics that focus on geometry. It can also be shown in relation to straight lines, polygons, and angles. When considered together, all of these facts are characteristics of the circle. Now that we know what a circle is, let us look at some of its basic qualities to gain a better understanding of it.
Math Facts
Know more about other concepts of Geometry on what is perimeter, Geometric shapes , Perimeter of circle, herons formula and Area of triangle.
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