Can you work out the ratio of the area of the equilateral triangle to the area of the regular hexagon shown, assuming both have perimeters of the same length?

**Answer**: To get to this week’s answer, 3:2 (or 1.5), readers came up with an impressive range of solutions (see below). Very well done, in order of reply, to: **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **Flemming Vorbeck**, purchasing manager commercial, NDI Group A/S, Brørup, Denmark; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **France Veillette**, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; **Michael Easton**, director, Globus Group, Trafford Park, Manchester, UK; and everyone else who had a go.

Solutions

*Stephan Paischer*

The answer is: area B of hexagon = 1.5 x area A of triangle

Triangle A with area A and side a

Hexagon B with area B and side b

Perimeter triangle: 3a -> equals to

Perimeter hexagon: 6 b

So: 3 a= 6 b

a = 2b

Area triangle:

A = (a^2)/4 x sqrt(3)

Area hexagon:

B = 6 x (b^2)/4 x sqrt(3)

The hexagon is made up of 6 equal-sided triangle with side length b

Remodel area triangle by inserting a = 2b to

A = 4 x (b^2) /4 x sqrt(3)

(b^2) = a / sqrt(3)

Insert into hexagon formula:

B = 3/2 x A / sqrt(3) x sqrt(3)

B = 3/2 x A

B = 1.5 x A

*John Bowen*

Area of an equilateral triangle is given by [Sq Rt 3* [D sqd]/4 where D = length of side.

So for our Triangle, D = L/3 where L is perimeter

So the area of our triangle is [Sq Rt 3] *[L/3] Sqd /4 = [Sq Rt 3] * L Sqd/36.

For our hexagon, this is made up of 6 equilateral triangles each of side length L/6

So the total area is 6 * [ Sq Rt 3 * {L/6} Sqd}]/4 which simplifies to 1.5 [Sq Rt 3] *L Sqd/36

So the area of the hexagon is 1.5 times the area of the triangle with the same perimeter.

*Flemming Vorbeck*

To work in actual numbers, let’s start by setting the perimeter to a given length, let’s say 30. Unit not important to solve the task.

Each side (s) in the two shapes will then be:

Triangle: 30/3 = 10; Hexagon: 30/6 = 5

The area of an equatorial triangle is calculated as follows:

A = √3/4 s^2

This gives our triangle with a perimeter of 30 the following area:

√3 = 1.732051

1,732051/4 = 0.433013

s^2 = 10^2 = 100

Triangle area: 0.433013 x 100 = 43.30127

The area of a hexagon is calculated as follows:

A = (3√3 s^2)/2

This gives our hexagon also with a perimeter of 30 the following area:

√3 = 1.732051

s^2 = 5^2 = 25

Hexagon area: 3 x 1.732051 x 25 divided by 2 = 129.9038106/2 = 64.95191

The ratio between the area of the triangle and the hexagon would then be 1 to (64.95191)/(43.30127) = 1.5

*Andrew Knox*

Answer: Ratio of Area Hexagon to Area Triangle is 6 : 4 (or 1.5).

Assuming both are regular polygons, then if the sides of hexagon are all of length b, then the sides of the triangle (if the perimeters are of equal length) will each be of length 2b.

The triangle can be divided into 4 four equal equilateral triangles each with sides of length b, by drawing lines from the centre of each side to the centre of the adjacent sides.

The hexagon can be divided into 6 equilateral triangles by slicing it up like a cake from each corner to the centre. Each of these 6 equilateral triangles will also have sides of length b.

So the ratio of the area of the hexagon to the area of the triangle is simply 6:4, or 1.5.

I must admit that by resorting to published formula's (and without going back to first principles of sine/cosine, internal angles etc., which one ought to do🙈), I got the answer wrong, so this time I have gone for the stylish answer😎.

*Michele Girardi *

The ratio of the area of the triangle to the one of the hexagon is 2/3 .

This can be seen graphically by :

- subdiving the triangle in 4 units by uniting the middle points of the sides

- subdividing the hexagon in 6 units by uniting the vertexes

- veryfing that the subtriangles have the same area , so the ratio is 4/6 = 2/3

Alternatively :

- the area of a polygon is given by perimeter x apothem / 2

- since the triangle and the hexagon have the same perimeter, the ratio in given by the ratio of the apothems

- setting the side of triangle = 1, it's ap.Tria. = 1/2*1*tan 30° = 1/2*sq.root(3) /3 ap. Hex = 1/4 *1*tan 60°= 1/4*sq.root(3) so area ratio = ap.Tria/ ap.Hex = 2/3.

*France Veillette*

The ratio of the area of the triangle to the area of the hexagon is 2/3.

Given that the perimeter is the same, one side of the triangle is double one segment of the hexagon. Using the formula for the area of a triangle being side2*3/4 and the area of the hexagon being 3*3*segment2/2; the ratio triangle/hexagon is 2/3.

New teaser on Monday.