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?
- A
- B
- C
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.
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?
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. 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. 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 are very important numbers to keep in mind)
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?