Question

Find ratio if perimeter of an equilateral triangle and a regular hexagon are equal.

Answer 1

The perimeter of an equilateral triangle is equal to the perimeter of a hexagon. What is the ratio of their areas?
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Meher Prudhvi
Meher Prudhvi
2 Answers in Mathematics
Soham Mukherjee
Soham Mukherjee, Higher Secondary Science, South Point High School, Kolkata (2019)
4 Answers in Mathematics
Sanjeev Dhankhar
Sanjeev Dhankhar
2 Answers in Geometry
Atmik PandeyEr Sandeep Singh TomarABHISHEK SIROHIView More or Search
3 ANSWERS
Anirban Ghoshal
Anirban Ghoshal, Amateur. Had some math in college as part of a B. Tech (IT) degree.
Answered Sep 16, 2015
This is a regular hexagon that's been broken down into 6 equilateral triangles (congruent, of course):

The sum of any three sides of this hexagon is equal to the perimeter of one of the triangles. The area of the hexagon is 6 times the area of one of the triangles.

Now, an equilateral triangle whose sides are twice as long as those shown above will have its perimeter equal to the hexagon. This new equilateral triangle's area will then be 4 times that of each triangle shown above. That's 3/2 the area of the hexagon.
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Leela Prasad
Leela Prasad
Answered Jan 5
Let the side of an equilateral triangle be a and that of a hexagon be a’.

Now:-

Perimeter of an equilateral triangle=3a;

Perimeter of hexagon=6a’;

Given that both Perimeters are equal:therefore

3a=6a’ => a=2a’;

If the side of equilateral triangle is known, then its area is given by √3÷4 times the square of side.

SO area of triangle is (√3/4)*a*a; here substituting a with a’ we get its area as (√3/4)*4*a’*a’; ————-(1)

Hexagon can be divided into six equal hexagons and the total area is

6*(√3/4)*a’*a’—————-(2)

Taking ratio for (1) and (2) you will get 2/3.