Areas of Sectors formed at the Vertices of a Triangle

Areas of Sectors formed at the Vertices of a Triangle

Objective
To verify that sum of areas of three sectors of the same radii ‘r’ formed at the vertices of any triangle is          πr2 by paper cutting and pasting.

Prerequisite Knowledge

1. Concept of different types of triangles.
2. Definition of a sector.
3. Area of circle = πr2, r → radius.

Materials Required
Glazed paper, sketch pens, fevicol, a pair of scissors, pencil, geometry box.

Procedure

1. Draw three different types of triangles on a glazed paper as shown in fig. (i).
   (i) Equilateral ∆ABC
   (ii) Isosceles ∆PQR
   (iii) Scalene ∆XYZ
   
2. Cut an equilateral ∆ABC as shown in fig.(ii)
   
3. Taking vertices A, B and C as centres of ∆ABC, draw three sectors of same radii r.
   
4. Cut these three sectors and marked them as 1,2,3 and fill different colours.
   
5. Draw a straight line and mark any point ‘O’ on it. Place three sectors 1,2, 3 adjacent to each other so that the vertices A, B, C coincide with ‘O’ without leaving any gap as shown in fig. (v).
   
6. The same process (steps 1-5) can be taken up with isosceles triangle and scalene triangle fig. (i).

Observation
The shape formed on the straight line is a semi circle
∴ area of circle = πr2
∴ area of semicircle =  πr2

Result
It is verified that the sum of areas of three sectors of same radii ‘r’ formed at the vertices of any triangle is  πr2.