Areas Related to Circles

Areas Related to Circles

1.A circle is a set of points in a plane that are at an equal distance from a fixed point. The fixed point is called the centre of circle and equal distance is called the radius of the circle.

2.A line segment joining the centre of the circle to a point on the circle is called its radius.

3.A line segment joining any two points of a circle is called a chord. A chord passing through the centre

4.of circle is called its diameter.

5.The distance around the boundary of the circle is called the perimeter or the circumference of the circle.

6.Circumference (perimeter) of a circle = $\pi d$ or $2\pi r$, where d is he diameter, r is the radius of the circle and $\pi =\frac{22}{7}$

7.Perimeter of a semi circle or protractor = $\pi r+2r$

8.Perimeter of a quadrant = $\frac{1}{4}Circumference+2r=\frac{\pi r}{2}+2r$

9.Distance moved by a wheel in 1 revolution = Circumference of the wheel.

Number of revolutions in one minute $=\frac{Distancemovedin1minute}{Circumference}$

10.The region enclosed inside a circle is called its area.

11.Area of a circle = ${\pi r}^2$

12.Area of a semi circle$=\frac{1}{2}{\pi r}^2$

13.Area of a quadrant = $\frac{1}{4}$ Area of circle$=\frac{1}{4}{\pi r}^2$

14.Circles having the same centre but different radii are called concentric circles.

Area enclosed by two concentric circles = ${\pi R}^2-{\pi r}^2=\pi (R^2-r^2)=\pi (R+r)(R-r)$

Where, R and r are radii of two concentric circles

15.The part of the circumference between the two end points of the chord is called an arc. In the figure, arc $\widehat{AB}$ is shown.

16.A diameter of circle divides a circle into two equal arcs, each known as a semi-circle.

17.An arc of a circle whose length is less than that of a semicircle of the same circle is called a minor arc.

18.An arc of a circle whose length is greater than that of a semicircle of the same circle is called a major arc.

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.

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, the 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

All these formulas are tabulated as given below for quick revision.

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.

Area Of Shapes

In Geometry, a shape is defined as the figure closed by the boundary. The boundary is created by the combination of lines, points and curves. Basically, there are two different types of geometric shapes such as:

Two – Dimensional Shapes

Three – Dimensional Shapes

Each and every shape in the Geometry can be measured using different measures such as area, volume, surface area, perimeter and so on. In this article, let us discuss the area of shapes for 2D figures and 3D figures with formulas.

2D shapes

The two-dimensional shapes (2D shapes) are also known as flat shapes, are the shapes having two dimensions only. It has length and breadth. It does not have thickness. The two different measures used for measuring the flat shapes are area and the perimeter. Two-dimensional shapes are the shapes that can be drawn on the piece of paper. Some of the examples of 2D shapes are square, rectangle, circle, triangle and so on.

Area of 2D Shapes Formula

In general, the area of shapes can be defined as the amount of paint required to cover the surface with a single coat. Following are the ways to calculate area based on the number of sides that exist in the shape, as illustrated below in the fig.

Let us write the formulas for all the different types of shapes in a tabular form.

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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

Important Questions

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Multiple Choice questions-

1. Perimeter of a sector of a circle whose central angle is 90° and radius 7 cm is

(a) 35 cm

(b) 25 cm

(c) 77 cm

(d) 7 cm

2. The area of a circle that can be inscribed in a square of side 10 cm is

(a) 40π cm²

(b) 30π cm²

(c) 100π cm²

(d) 25π cm²

3.The perimeter of a square circumscribing a circle of radius a units is

(a) 2 units

(b) 4α units

(c) 8α units

(d) 16α units

4. The perimeter of the sector with radius 10.5 cm and sector angle 60° is

(a) 32 cm

(b) 23 cm

(c) 41 cm

(d) 11 cm

5. In a circle of diameter 42 cm, if an arc subtends an angle of 60° at the centre, where π = 227 then length of arc is:

(a) 11 cm

(b) 227 cm

(c) 22 cm

(d) 44 cm

6. The perimeter of a sector of radius 5.2 cm is 16.4 cm, the area of the sector is

(a) 31.2 cm²

(b) 15 cm²

(c) 15.6 cm²

(d) 16.6 cm²

7. If the perimeter of a semicircular protractor is 72 cm where π = 227, then the diameter of protractor is:

(a) 14 cm

(b) 33 cm

(c) 28 cm

(d) 42 cm

8. If the radius of a circle is doubled, its area becomes

(a) 2 times

(b) 4 times

(c) 8 times

(d) 16 times

9. If the sum of the circumferences of two circles with radii R₁ and R₂ is equal to circumference of a circle of radius R, then

(a) R₁ + R₂ = R

(b) R₁ + R₂ > R

(c) R₁ + R₂ < R

(d) Can’t say.

10. The perimeter of a circular and square fields are equal. If the area of the square field is 484 m² then the diameter of the circular field is

(a) 14 m

(b) 21 m

(c) 28 m

(d) 7 m

  Very Short Questions:

Long Questions :

Assertion Reason Questions-