How to draw a pentagon of an exact height?

Question

Here are the criteria: A pentagon is made up of 5 isoceles triangles. I need the length of the height, base, and isoceletic side of one of these triangles. The height of the total pentagon must be equal to 12 inches. In other words, if you draw it out, one isoceletic side plus the height of one of these triangles must be equal to 12. I need this to be exact, so no drawing out 5 lines 72 degrees aprt from each other and going out until the height is correct or anything like that. I think this will be a fun challenege for some of the math buffs out there. First correct answers that I understand (and I should, I think im pretty smart) will get best answer. Enjoy and thank you!

Answer 1

Okay, a circle has 360 degrees. Since a pentagon has five of these isosceles triangles that meet in the center of the pentagon, each central angle is 360/5 = 72 degrees.

This means the other two angles of each triangle are (180-72)/2 = 54 degrees.

Now cut an isosceles triangle in half. This gives us two triangles with angles 90-54-36.

Let x be the length of one side of the pentagon. Then the side opposite the 36 degree angle (of these 90-54-36 triangles) will be length 0.5x. Let y be the length of the triangle leg adjacent to the 36 degree angle. We can now calculate y:
tan 36 = 0.5x / y
y = 0.5x / tan 36
y = 0.688x
This is also the height of each isosceles triangle.

The hypotenuse of the 90-54-36 triangle can be calculated too:
sin 36 = 0.5x / hypotenuse
hypotenuse = 0.5x / sin 36
hypotenuse = 0.851x

The height of the pentagon equals 0.851x + 0.688x, or 1.539 x.
We want 1.539x to equal 12 inches.
Hence, x = 12 inches / 1.539, or approximately 7.8 inches.

Each 54-72-72 triangle will have one side of length 7.8 inches, two sides of length (0.851*7.8= ) 6.6 inches, and a height of (0.688 * 7.8 = ) 5.4 inches.

Answer 2

I think the best way to construct the pentagon of height 12 inches is construct it using a compass and straightedge in a circle. Its easy to look up a method for doing so on the internet. The trick is to determine what the radius of the circle should be so the the pentagon has a height of 12 inches.

Let
r = radius circle
a = apothem of inscribed pentagon
h = height inscribed pentagon

h = r + a

The central angle of a pentagon is 360/5 = 72°.

cos36° = (√5 + 1)/4

cos(72/2) = cos36° = a/r

a = rcos36°

h = r + a = r + rcos36° = r(1 + cos36°) = r[1 + (√5 + 1)/4]

h = r[1 + (√5 + 1)/4] = r[(5 + √5)/4]

12 = r[(5 + √5)/4]

r = 12 / [(5 + √5)/4] = 48 / (5 + √5)

r = 48(5 - √5) / [(5 + √5)(5 - √5)] = 48(5 - √5) / (25 - 5)

r = 48(5 - √5) / 20 = 12(5 - √5) / 5 ≈ 6.6334369 inches

If you construct a pentagon inside a circle with radius

12(5 - √5) / 5 inches

you will get a pentagon exacly 12 inches high.

Answer 3

How To Draw A Pentagon

Answer 4

Let side be y, triangle height = h, base be 2x

Then y + h = 12
Also, cos(36) = h/y
Hence, h = ycos(36)

y(1 + cos(36)) = 12
y = 6.63

So h = 5.367

x/y = sin(36)
x = 3.90
Base = 7.8

Answer 5

You will need a: protractor, compass, ruler and a sharp pencil.