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