Areas of Regular Polygons

Part 1: Let's Experiment a Little! Off the top of your head what is the area of each of these shapes.

Off the top of your head what should be the area of this triangle

Off the top of your head what should be the area of this square

How about a hexagon?

NOPE No NOOOOOOOOO

Part 2: Unfortunately finding the area of a regular hexagon is not as simple as finding the area of other shapes. But lets take it one step at a time.

We know how to find the area of a square, the area of a rectangle, and the area of a triangle so lets use one of those shapes to fill in the hexagon below.

Use the polygon tool to fill in the hexagon using only one type of shape listed above.

Which type of polygon could fill in the entire hexagon?

Square Rectangle Triangle

Part 3: Finding our area would be a lot easier if all of our triangles were equally sized. Use the center given below and the polygon tool to make 6 equally sized triangles.

Now that we have our polygon broken up into triangular slices we now just have to find the specific measurements of those triangles.

First let's try to circumscribe the hexagon. Form a circle using the point G as your center and one of the corners of the hexagon as your radius.

Think about how many degrees of rotation a circle has, now think about how it is split by the triangles.[br][br]What do you think is the measurement of the angle shown above?

Now Measure the angle, was your initial assumption correct?

Part 4: Now that we know that each angle near the center has a measure of 60 degrees lets focus in on that triangle and figure out its area

If we are given the base of the triangle what else would we need to find the area of the triangle?

Use the perpendicular bisector tool to bisect the triangle.

Write down the angles of the newly made right triangle, on the left and right halves.[br][br]Also write down what the length of the side split by the bisector is.

Use the special right triangle above to get the height of the triangle. (hint: for a 30 , 60 , 90 the numbers [math]1,\sqrt{3},2[/math] are very important numbers to keep in mind)

You'll get a special right triangle in the above example but how would you find the altitude (the height) in a more general / less specific example?

Lets look at a general right triangle.

Given angle α and knowing the size of side "a" how would you find the size of side "o"? (Hint: Try to work out the problem by hand using trig functions)

Now that we know the height of the triangle we can finally try to find the area of the triangle.

Use the base 4 and height 3.5 to find the area of the triangle.

Part 5: Now putting the triangle back into the hexagon if there are six triangles in total, what would be the area of the hexagon.

What is the area of the hexagon found above?

To summarize to find the area of any regular polygon with "n" amount of sides.

1. Split your polygon into "n" triangles.[br]2. Circumscribe the polygon.[br]3. Find the the angle of the triangle.[br]4. Perpendicularly bisect the triangle to find the altitude.[br]5. Calculate what the altitude / height of the triangle is by using trig functions.[br]6. Calculate the area using the entire side of the triangle (not the split side)[br]7. Multiply this area by the number of triangles your "n triangles"[br]