Question

the diagram shows two circles with radius r cm. the centre of each circle lies on the circumference of the circle. Find, in terms of r, the extract area of shaded region ​

Answer 1

$\setlength{\unitlength}{1cm}\begin{picture}(5,5)\multiput(0,0)(0.7,0){2}{\circle*{0.1}}\multiput(0.35,-0.6)(0,1.2){2}{\circle*{0.1}}\put(0,0){\circle{2}}\put(0.7,0){\circle{2}}\multiput(0,0)(0.35,-0.6){2}{\line(3,5){0.33}}\multiput(0.35,0.6)(-0.35,-0.6){2}{\line(3,-5){0.33}}\put(0,0){\line(1,0){0.7}}\put(0.2,0.1){ \tiny60\textdegree }\put(-0.25,0){ \tiny\text{A} }\put(0.25,0.7){ \tiny\text{B} }\put(0.75,0){ \tiny\text{C} }\put(0.25,-0.85){ \tiny\text{D} }\end{picture}$

Here, the centers of the circle are taken as A and C. Let B and D be the points of intersection of the circle.

Triangles ABC and ACD are equilateral because AB = BC = CD = AD = r. Thus each angle equals 60°.

The sectors ABC, BCA, ADC and CDA have same area.

So, what I'm going to do is that,

• First the area of one of the sectors, say sector ABC, is found.

• Then the area of one of the equilateral triangle, say triangle ABC, is subtracted from double this area.

• Then the result is doubled.

This gives our answer, the area of the shaded region in the diagram.

Let's find it!

Area of sector ABC = πr² × 60° / 360° = πr²/6

Area of triangle ABC = r²√3/4

Hence the answer is,

2(2πr²/6 - r²√3/4) = r²(2π/3 - √3/2)

Taking π = 3.142 and √3 = 1.732,

r²(2 × 3.142 / 3  -  1.732 / 2) = 1.229r²

Answer 2

Step-by-step explanation:

setlength{\unitlength}{1cm}\begin{picture}(5,5)\multiput(0,0)(0.7,0){2}{\circle*{0.1}}\multiput(0.35,-0.6)(0,1.2){2}{\circle*{0.1}}\put(0,0){\circle{2}}\put(0.7,0){\circle{2}}\multiput(0,0)(0.35,-0.6){2}{\line(3,5){0.33}}\multiput(0.35,0.6)(-0.35,-0.6){2}{\line(3,-5){0.33}}\put(0,0){\line(1,0){0.7}}\put(0.2,0.1){$\tiny\text{$60\textdegree$}$}\put(-0.25,0){$\tiny\text{A}$}\put(0.25,0.7){$\tiny\text{B}$}\put(0.75,0){$\tiny\text{C}$}\put(0.25,-0.85){$\tiny\text{D}$}\end{picture}

Here, the centers of the circle are taken as A and C. Let B and D be the points of intersection of the circle.

Triangles ABC and ACD are equilateral because AB = BC = CD = AD = r. Thus each angle equals 60°.

The sectors ABC, BCA, ADC and CDA have same area.

So, what I'm going to do is that,

First the area of one of the sectors, say sector ABC, is found.

Then the area of one of the equilateral triangle, say triangle ABC, is subtracted from double this area.

Then the result is doubled.

This gives our answer, the area of the shaded region in the diagram.

Let's find it!

Area of sector ABC = πr² × 60° / 360° = πr²/6

Area of triangle ABC = r²√3/4

Hence the answer is,

2(2πr²/6 - r²√3/4) = r²(2π/3 - √3/2)

Taking π = 3.142 and √3 = 1.732,

r²(2 × 3.142 / 3 - 1.732 / 2) = 1.229r².