Areas of Regular Polygons

1
Areas of Regular Polygons Lesson 11.5
2
Equilateral Triangle
Remember drop an altitude and you create two
30-60-90 triangles.
What is the measure of the sides and altitude in
terms of one side equaling s? Altitude sv3
2
3
C
Given ? CAT is equilateral, and TA s Find the
area of ?CAT
A
T
S
?A?CAT
2
4
Theorem 106 Area of an equilateral triangle
the product of 1/4 the square of a side and the
square root of 3. Where s is the length of a side
Aeq?
2
5
An equilateral triangle has a side of 10 cm long.
Find the area of the triangle. A 102(v3)
4 A 25v3 cm2
6
Area of a regular polygon Remember all interior
angles are congruent and all sides are equal.
N
Regular pentagon O is the center OA the
radius OM is an apothem
T
E
O
A
M
P
7
You can make 5 isosceles triangles in a pentagon.
Any regular polygon Radius is a segment joining
the center to any vertex Apothem is a segment
joining the center to the midpoint of any side.
8

• Apothems
• All apothems of a regular polygon are congruent.
• Only regular polygons have apothems.
• An apothem is a radius of a circle inscribed in
  the polygon.
• An apothem is the perpendicular bisector of a
  side.
• A radius of a regular polygon is a radius of a
  circle circumscribed about the polygon.
• A radius of a regular polygon bisects an angle of
  the polygon.

9
Theorem 107 Areg poly ½ ap Area of a regular
polygon equals one-half the product of the
apothem and the perimeter. Where a apothem p
perimeter
10
A regular polygon has a perimeter of 40 cm and an
apothem of 5 cm. Find the polygons area. A
½ap ½(5)(40) 100 cm2
11

• Find the area of a regular hexagon whose sides
  are 18 cm long.
• Draw the picture
• Write the formula
• Plug in the numbers
• Solve and label units

12
Find the perimeter Find each angle Find the
apothem
18cm
P 18(6) 108 cm Angles 720º/6 angles 120º
per angle Radius breaks it into 60º
angles. 30-60-90 triangle, apothem 9v3 cm
Write the formula, and solve.
13
A ½ ap A ½ (9v3)108 A 486v3 cm 2
14
Team Challenge A square is inscribed in an
equilateral triangle as shown. Find the area of
the shaded region.
15
2x xv3 12 x 12 2 v3 x 12(2
v3) A (shaded) ½ (12)(6v3) 12(2
v3)v32 1764v3 - 3024
xv3
x
x
xv3