Introduction of Areas Related to Circles

Area of a Circle

Area of a circle is πr2, where π=22/7 or ≈3.14 (can be used interchangeably for problem-solving purposes)and r is the radius of the circle.
π is the ratio of the circumference of a circle to its diameter.

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Circumference of a Circle

The perimeter of a circle is the distance covered by going around its boundary once. The perimeter of a circle has a special name: Circumference, which is π times the diameter which is given by the formula 2πr

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Segment of a Circle

A circular segment  is a region of a circle which is “cut off” from the rest of the circle by a secant or a chord.

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Sector of a Circle

A circle sector/ sector of a circle is defined as the region of a circle enclosed by an arc and two radii. The smaller area is called the minor sector and the larger area is called the major sector.

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Angle of a Sector

The angle of a sector is that angle which is enclosed between the two radii of the sector.

Length of an arc of a sector

The length of the arc of a sector can be found by using the expression for the circumference of a circle and the angle of the sector, using the following formula:

L= (θ/360°)×2πr

Where θ is the angle of sector and r is the radius of the circle.

Area of a Sector of a Circle

Area of a sector is given by

(θ/360°)×πr2

where ∠θ is the angle of this sector(minor sector in the following case) and r is its radius

Area of a sector

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Area of a Triangle

The Area of a triangle is,
Area=(1/2)×base×height
If the triangle is an equilateral then
Area=(√3/4)×a2  where “a” is the side length of the triangle.

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Area of a Segment of a Circle

Area of segment APB (highlighted in yellow)
= (Area of sector OAPB) – (Area of triangle AOB)

=[(∅/360°)×πr2] – [(1/2)×AB×OM]

[To find the area of triangle AOB, use trigonometric ratios to find OM (height) and AB (base)]

Also, Area of segment APB can be calculated directly if the angle of the sector is known using the following formula.

 =[(θ/360°)×πr2] – [r2×sin θ/2 × cosθ/2]

Where θ is the angle of the sector and r is the radius of the circle

Visualizations

Areas of different plane figures

–    Area of a square (side l) =l2
–    Area of a rectangle =l×b, where l and b are the length and breadth of the rectangle
–    Area of a parallelogram =b×h, where “b” is the base and “h” is the perpendicular height.

parallelogram

Area of a trapezium =[(a+b)×h]/2,

where

a & b are the length of the parallel sides

h is the trapezium height

Area of a rhombus =pq/2, where p & q are the diagonals.

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Areas of Combination of Plane figures

For example: Find the area of the shaded part in the following figure: Given the ABCD is a square of side 28 cm and has four equal circles enclosed within.

Area of the shaded region

Looking at the figure we can visualize that the required shaded area = A(square ABCD) − 4 ×A(Circle).

Also, the diameter of each circle is 14 cm.
=(l2)−4×(πr2)
=(282)−[4×(π×49)]
=784−[4×22/7×49]
=784−616
=168cm2