Find the area of the circle outside the triangle

Question

Three cicles are drawn, with the vertices of  an equilateral trianglt as centres , so that each touches the remaining other two. If the perimeter of the triangle is 84cm; calculate:-

(i) the area of the portion of each circle outside the triangle .

(ii) the area of that portion of the triangle which is not included in any of the circles drawn.

Answer 1

The sides of the triangle are the same length so one side=84/3=28cm. The circles touch other and they are of equal size, so the radius of each is half the side of the triangle, 14cm.

(i) The angle of the triangle is 60 degrees which is 1/6 of 360 so the area outside the triangle is 5/6 of the area of the circle=5/6 of 196π where 196=14^2. The area is therefore 490π/3=513.1268 sq cm approx.

(ii) The area of the triangle is 1/2base*height, and the height is given by Pythagoras as √(4r^2-r^2)=r√3 where r=14cm, so the area is 14*14√3=339.482 approx. From this we need to subtract the area of 3 circle segments, each being 1/6 of the area of a circle: 196π/6=98π/3. Therefore we need to subtract 98π=307.8761. The area of the centre of the triangle is therefore 196√3-98π=31.61 sq cm approx.