Formula for the area of a regular polygon

Geometry G5-3 Investigation Name _______________________ Per _____

Area of regular polygons

So far we have learned these area formulas:Area of a rectangle = base x heightArea of a parallelogram = base x heightArea of a triangle = (base x height)

Area of a trapezoid = h(base1 + base2)

Area of a kite = (diagonal1)(diagonal2)

Area of a regular polygon:

Step 1: A regular polygon has all sides the same length. Label the sides s below.

Step 2: Every regular polygon can be perfectly inscribed in a circle (as shown below). This allows us to cut the polygon into triangles from the center of the polygon to the vertices. What part of the circle did we draw? ____________

Are these triangles congruent? ________ How do you know? ________

Step 3: Since the triangles are congruent, we will focus on the area of one triangle.

Draw the height of one of the triangles in each shape. This is called the apothem of the polygon. Label them a.

PentagonHexagon

Step 4: Find the area of one triangle.

What is the base of our triangles? _____ What is the height? ______

So Area = _____________

Step 5: Find the area of the entire regular polygon:

How many triangles are there? Pentagon: ______ Hexagon: _____

Multiply the area of 1 triangle, by the number of triangles:

Area of a regular pentagon = __________ Area of a regular hexagon = __________

Using this information we can find the formula for the area of ANY regular polygon.

number of sides

5

6

7

8

n

Area of the regular polygon with side length s

Perimeter

If you were given the perimeter, P, instead of a side length, what would be the formula for the area of the regular polygon?

Find the area of each regular polygon below:

1.2.

8 in.

5 m 10 in.

8 m.

3.A regular polygon with a perimeter of 20 cm. and apothem of 3 cm.

4.5.

3 in.

12 in.

7 ft. 6 ft.