# Thread: how to create any shape from an area

1. ## how to create any shape from an area

Hey all,

I usually don't deal with heavy maths on a regular basis so the way I communicate this may not be spot on, but here goes.

I wish to create shapes of ANY number of sides that have a specific area and I know the area of which they have to be.

For example, the area is 9cm squared, so to create a square shape from that I would squareroot it to find the length and breadth in order to draw the shape. I can also manage to do the same for a triangle by reversing (1/2(bxh)), but the shapes I want to draw are:

Semicircle - (I think i can do this one)
Heptagon - 7sides
dodecagon - 12 sides
icosagon - 20 sides

Any ideas how I create these shapes that have specific areas?

Thanks very much in advance,
Dave

2. There is a formula you can use for the area of a regular polygon.

$Area = \frac{n}{2}\cdot{(radius)^{2}}\cdot{sin(\frac{2\pi }{n})}$. Where n is the number of sides.

For instance, for the septagon(7 sides), say you want an area of 20 square units:

$\frac{7}{2}r^{2}sin(\frac{2\pi}{7})=20$

Solve the radius and we get $r\approx{2.703}$

Then, a side length would be $2rsin(\frac{\pi}{n})=2.703\cdot{2sin(\frac{\pi}{7} )}=2.346$

I've known these formulas for sometime, but here's a link to a useful site.

Regular Polygons - Properties

3. Here is another link that may help.
Regular Polygon -- from Wolfram MathWorld

4. ## excellent

excellent thanks.

Not being fluent with advanced maths, how would I type this into a calculator to work it out. (step by step)

I have no idea what r2 is.

Thanks.
Dave

5. I mentioned it in my first post. r is the radius, so r^2 is the radius squared.

6. ahh right, now I see.

Thanks alot.
Topic closed I guess.

7. Originally Posted by plastiman
excellent thanks.

Not being fluent with advanced maths, how would I type this into a calculator to work it out. (step by step)

I have no idea what r2 is.

Thanks.
Dave
This is not advanced math, just some trig and algebra. Though, I suppose it may seem that way.

Some calculators have solver functions. Such as a TI-89, TI-92, Voyage 200, etc. To do it algebraically, the old-fashioned way we may proceed as such:

$\frac{7}{2}r^{2}sin(\frac{2\pi}{7})=20$

Multiply both sides by 2/7:

$r^{2}sin(\frac{2\pi}{7})=\frac{40}{7}$

Divide by $sin(\frac{2\pi}{7})$:

$r^{2}=\frac{40}{7sin(\frac{2\pi}{7})}$

Square root of both sides:

$r=\sqrt{\frac{40}{7sin(\frac{2\pi}{7})}}\approx{2. 703}$

See?.

Here's a screen capture from the solve function in my TI-92.

As you can see, it easily solves for r.

It gives two solutions, but only one is viable because the other is negative.