[bonanova]

An equilateral triangle and a regular hexagon have the same perimeter.

Find the ratio of their areas.

Simple enough.

Can you find three different approaches/solutiions?

[Guest]

2:3. I'll give the way I solved it if I'm correct.

[plainglazed]

for the perimeter to be the same the hexagon has side length of 1/2 the equilateral triangle. if you bisect each side of the triangle and connect those points, you have four equilateral triangles of side length 1/2 of the original triangle. if you connect the vertices of the hexigon with the center point, you have six equilateral triangles of side length 1/2 of the original triangle. so the ratio of the area of the equilateral triangle to a hexagon of equal perimeter is 2:3

the area of an equilateral triangle is {sqrt(3)/4}s^2 and the area of a hexagon is {3sqrt(3)/2}s^2 where s is the side length of each. so substituting the s/2 as the side length of the hexagon and defining the ratio of the triangle area to the hexagon area as x you get: x{sqrt(3)/4}s^2={3sqrt(3)/2}s^2/4 you get x=3/2 or the area of the triangle is 2/3 the area of the hexagon.

still thinking about a third...

EDIT:

by inspection, half the hexagon is 3/4 of the triangle so twice that would be 3/2. pretty similar to the first way above tho.

Edited May 7, 2011 by plainglazed

[Guest]

At first I used Plaingazed's third way, but just to make sure, I then used his 2nd way

[Guest]

The hexagon has 1/2 the length of the edge and it can be divided into 6 triangles, so one triangle with edge 2a vs 6 triangles with edge a, since the area is edge² then the ratio is 4x1 : 1x6, which is 2:3

Edited May 7, 2011 by Anza Power

[curr3nt]

Let us make the perimeter equal to 6x.

Sides of the triangle will then be 2x and the sides of the hexagon will be x.

The area of the triangle is height * base / 2.

h = sqrt((2x)^2 - x^2) = sqrt(3x^2) = sqrt(3)x

So the area of the triangle is sqrt(3)x * 2x / 2 = sqrt(3)x^2

The area of the hexagon is 6 * the area of equilateral triangle with the side of x.

h = sqrt(x^2 - (x/2)^2) = sqrt(3x^2/4) = sqrt(3)x/2

Area of one of those triangles is sqrt(3)x/2 * x / 2 = sqrt(3)x^2/4

Area of the hexagon would be 6 * sqrt(3)x^2/4 = 3/2 * sqrt(3)x^2

So sqrt(3)x^2 vs 3/2 * sqrt(3)x^2

Or 2:3 for Triangle:Hexagon

[Guest]

If you divide the two into equilateral triangles each with side 1/2 the length of original triangle, you get 4 small triangles inside the big triangle by connecting mid points of sides and 6 small triangles inside the hexagon by connecting each edge to the centre. So the ratio of areas is 2/3

Edited May 9, 2011 by DeeGee

[bonanova]

Great job all.

Here are some approaches that occurred to me when I posted:

I think they all were covered.

Area formulas:

Hexagon with side s: A_h = (3/2)(SQRT[3])s²

Triangle with side 2s: A_t[2s] = (SQRT[3])s² = (2/3) A_h

And some ways that don't use area formulas:

Make a triangle from a hexagon by unfolding:

This doubles half of both the area and the perimeter.

A_t = 3/2 A_h

P_t = 3/2 P_h

Scaling [linear] the triangle down by 3/2 to equate perimeters gives

A_t = 2/3 A_h

Make a hexagon from 6 triangles:

A_h = 6 A_t

P_h = 2 P_t

Scaling [linear] down the hexagon by 2 to equate perimeters gives

A_h = 3/2 A_t

Make equal-perimeter triangle and hexagon from triangular building blocks:

Count the triangles:

A_t = 4

A_h = 6 = 3/2 A_t.