Question

Find the ratio of length if an equilateral triangle and a regular hexagon which are on the circumference of the circle?

Answer 1

Answer:

√3

Step-by-step explanation:

Find the ratio of length if an equilateral triangle and a regular hexagon which are on the circumference of the circle?

Area of equilateral triangle = (√3 / 4)a²

a = side of equilateral triangle

center of circle will divide this triangle into 3 equal area triangle

area of any one triangle = (√3 / 4)a²/3

Triangle thus formed has one side = a

let draw perpendicular from center at side = P

Area = (1/2)aP

(1/2)aP =  (√3 / 4)a²/3

P = (√3 / 2)a/3

radius² = (a/2)² + P²

radius² =  a²/4 + a²/12

radius² =  4a²/12

radius² =  a²/3

a² = 3 radius²

a = √3 Radius

side of equilateral triangle inside circle = √3 Radius

Area of regular hexagon = 3√3 a²/2

a = side of hexagon

Hexagon can be divided into 6 equal area triangles

area of one triangle = √3 a²/4

let draw perpendicular from center at side = P

Area = (1/2)aP

(1/2)aP =  (√3)a²/4

P = (√3 )a/2

radius² = (a/2)² + P²

radius² =  a²/4 + 3a²/4

radius² =  4a²/4

radius² =  a²

a² = radius²

a =  Radius

side of hexagon inside circle = radius

side of equilateral triangle/side of hexagon inside circle = √3