Areas of Combinations of Plane Figures

Example

Find the area of the coloured part if the given triangle is equilateral and its area is 17320.5 cm2. Three circles are made by taking the vertex of the triangles as the centre of the circle and the radius of the circle is the half of the length of the side of the triangle.(\pi=3.14\ \text{and}\ \sqrt3=1.73205)

Solution

Area of shaded region = Area of ∆ABC – Area of 3 sectors

Area of ∆ABC = 17320.5 cm2

\dfrac{\sqrt3}{4}\times(\text{side})^2=1.7320.5

(\text{side})^2=\dfrac{17320.5\times4}{1.73205}=4\times10^4

Side = 200 cm

As the radius of the circle is half of the length of the triangle, so

Radius = 100 cm

Area of sector =\dfrac{\theta}{360^o}\times\pi r^2

=\dfrac{60^o}{360^o}\times\pi(100)^2

=\dfrac{1}{6}\times3.14\times(100)^2=\dfrac{15700}{3}\ cm^2

Area of  3 Sectors  =3\times\dfrac{15700}{3}\ cm^2

Area of shaded region = Area of ∆ABC – Area of 3 sectors

=17320.5-15700\ cm^2

=1620.5\ cm^2

IMPORTANT POINTS

(i) If two circles touch internally, then the distance between their centres is equal to the difference of their radii.

(ii) If two circles touch externally, then the distance between their centres is equal to the sum of their radii.

(iii) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.

(iv) The number of revolutions completed by a rotating wheel in one minute Distance moved in one minute

= Distance moved in one minute / circumference

EXAMPLE

The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?

SOLUTION

Radius of wheel =40\ cm

Circumference of wheel =2\ \pi r=2\ \pi\times40=80\pi\ cm

In one rotation, wheel covers distance equal to its circumference.

Distance covered by wheel in 10 minutes with speed of 66km/hr, is given by

\mathrm{Distance =Speed \times Time=(66\times1000\times100\ cm/60\ minutes)}\times10\ minutes

Or Distance travelled = 1100000\ cm

Number of Revolutions = Total Distance travelled by wheel / Distance travelled in one revolution

Or Number of revolutions =\dfrac{1100000\ cm}{80\ \pi\ cm}= 4375

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