Be it the wheels of a vehicle, the ripples that get formed when a raindrop hits a pond, cosmic bodies like planets, food items like pizzas and donuts, coins or even bangles, the circle has always existed in our lives since time immemorial. It has undeniably seeped into our day to day lives with its overwhelming versatility. The term circle finds its roots in the Greek word kirkos, which means hoop or circle. It also took influence from the Latin word circulus, which again means ring. It has fascinated mathematicians since ages and is a shape that is seen, studied, found and used beyond theory.

Understanding a circle in geometric terms:

Definition of Circle:

when a curved line having its endpoints joint to each other makes a round figure, all of whose points are equidistant from a fixed point or center, a circle is formed.

Parts of Circle:

• Chord:

It is a line segment whose endpoints lie on the circle itself.

• Tangent:

It is a line that touches the circle at exactly one point externally and is perpendicular to the radius.

• Radius:

It is the distance between the center of the circle and its boundary. It is half of the diameter.

• Diameter:

It is line which passes through the center of the circle and has its endpoints on the boundary of the circle. Diameter is twice of the radius.

• Circumference:

It is the perimeter of the circle.

• Center:

It is point which is exactly in the center of the circle, equidistant from every possible point on the boundary of the circle.

• Area:

The space enclosed by the boundary of the circle is termed as the area of the circle.

• Secant:

It is a line which intersects the circle in two points on the boundary.

• Arc:

A part of the circumference is referred to as an arc.

• Sector:

It refers to the region of the circle that’s bounded by an arc and two radii.

• Segment:

It is a portion of the circle that is bounded by a secant or a chord and an arc.

Major sector, Minor sector, Major arc, Minor arc, and Major Segment, Minor segment:

Circumference of a Circle:

The boundary of any closed geometric figure is referred to as its perimeter. Similarly, in the context of a circle, the perimeter is called as the circumference.

Circumference Formula

The formula for calculating the circumference of a circle = π x diameter = π x 2 x radius

π is read as Pi and its value is 3.14159265358....(rounded to 3.14)
Numerically, 22/7 is the closest rational number to π in terms of its value.
Pi is defined as the ratio of the circumference of a circle to its diameter. This means that no matter what the diameter of a circle, when we divide its circumference by its diameter, we will unfailingly obtain the value as Pi or π (a Greek alphabet).
π is also called Archimedes’ constant.

Formula for calculating the Measure of an arc:

Circles Example:

Q.1) If the radius of a circle is 7 cm, what will be the measure of its circumference?

Solution: We know that circumference = π x 2 x radius
= π x 2 x 7
= (22/7) x 2 x 7
= 44
∴ The circumference of the circle is 44 cm.

Circles Example:

Q.2) A walking track is in the shape of a ring whose inner circumference is 440 m and outer circumference is 616 m. Find the thickness of the track.

Solution: We know that circumference = = π x diameter
Let R and r be the outer and inner radii of ring.
Then 2πr = 440 …(inner ring)
2 × (22/7) x r = 440
∴ r = (440× 7)/(2 × 22)
∴ r = 70 m
and
2πR = 616 …(outer ring)
∴2 × (22/7) × R = 616
∴ R = (616 × 7)/(2 × 22)
∴ R = 98 m
Therefore, width of the track = (98 - 70) m = 28 m

Area of a Circle:

The area of a circle refers to the region enveloped by the circumference of the circle.

Area of a Circle Formula

The formula for calculating the area of a circle = π x radius²

Circles Example:

Q.3) If the circumference of a circle is 44 cm, find the area of the circle.

Solution: We know that circumference = π x 2 x radius
∴ 44 = 22 x 2 x radius/7
∴ 44 = 44 x radius/7
∴ radius = 7 cm
We know that area of a circle = π x radius²
= (22 x 7 x 7)/7
= 154 cm²

Formula for calculating the area of a sector:

Circles Example:

Q.4) A right angled triangle has been inscribed inside a circle, having its longest side as the diameter of the same circle. If the two smaller sides of the triangle measure 21 cm and 28 cm respectively, then what is the area of the circle?

Solution: We know that the angle inscribed in a circle is a right angle and the smaller sides of the triangle measure 21 cm and 28 cm.
∴ By Pythagoras’ theorem, AB² + AC² = BC²
∴ 21² +28² = BC²
∴ 441 + 784 = BC²
∴ 1225 = BC²
∴ √ 1225 = BC
∴ ± 35 = BC
Since the length of anything cannot be negative, BC = 35 cm
We know that BC is the diameter of the circle
Hence the radius = 35/2 cm
We know that the area of circle = π x radius²
= (22/7) x (35 x 35)/(2 x 2)
= 962.5 cm²

Circles Example:

Q.5) A dog has been tied to the corner of a square field. If the length of the rope with which it has been tied is 56 m and if the side of the field measures 100 m, then calculate the area of the field in which the dog cannot roam.

Solution: We know that all four angles of a square measure 90°.
We know that area of sector = θ × π x radius² / 360
∴ Area of the sector in which the dog can roam = 90x (22/7)x(56x56/360) = 2464 m²
We know that area of a square = side²
∴Area of the field = 100x100 = 10000 m²
∴The area of the field in which the dog cannot roam = Area of the field – Area of the sector in which the dog can roam.
= 10000 – 2464
= 7536 m²
Hence the area of the field in which the dog cannot roam = 7536 m²

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