Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
Objective: By the end of this lesson you will be able to find the area a regular polygon
Circumcircle:
Incircle:
Radius:
Apothem:
http://www.mathsisfun.com/definitions/apothem.html
Label the circumcircle, the incircle and a side. Draw and label 2 apothems and radii:
Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
Central angle of a polygon:
n=___
angle = ________
n=___
angle = ________
n=___
angle = ________
n=___
angle = ________
What conclusion can we make regarding the central angle of a polygon and its number of
sides?
Central angle of a regular polygon =
Find the central angle measure of each regular polygon
A. Heptagon
B. Octagon
C. Nonagon
Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
Let P be the center of the circle and the hexagon. Draw
PE, PD and PT and all radii. Mark congruencies.
1. What type of triangle is PED ?
2. Formulate a plan to find the area of the hexagon.
𝟏
Area of a small triangle= (𝟐 × 𝒂𝒑𝒐𝒕𝒉𝒆𝒎 ×
𝟏
𝒔𝒊𝒅𝒆
𝟐
Area of a regular polygon= 𝒏 (𝟐 × 𝒂𝒑𝒐𝒕𝒉𝒆𝒎 ×
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝒓𝒆𝒈𝒖𝒍𝒂𝒓 𝒑𝒐𝒍𝒚𝒈𝒐𝒏 =
)
𝒔𝒊𝒅𝒆
𝟐
)
𝟏
𝟏
× 𝑷𝒆𝒓𝒊𝒎𝒆𝒕𝒆𝒓 × 𝑨𝒑𝒐𝒕𝒉𝒆𝒎 = 𝑷𝒂
𝟐
𝟐
http://www.nasa.gov/mission_pages/cassini/multimedia/pia09187.html
Find the area of the storm if one side of the storm is 30 miles long.
Steps:
1- Find the perimeter of the hexagon
2- Find the central angle and the base angles.
Central Angle =
Base angle =
3- Draw the right triangle. Label the side,
apothem (altitude of the triangle)
4- Find the Area.
𝒔𝒊𝒅𝒆
𝟐
Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
Practice
1. Find the area of a regular hexagon whose perimeter is 48 inches.
Perimeter=
Central angle=
Base angle=
Side=
Draw your triangle here:
𝒔𝒊𝒅𝒆
Apothem=
Area=
2. Octagon with a perimeter of 80 inches and a radius of 10.45 inches.
3. Heptagon with a side of 3 and a radius of 3.45
𝟐
=
Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
4. Find the area of the shaded region,
if the radius of the circumcircle is 2 cm and
a length of the side is 2.352 cm.
Homework:
1. Find the area of each regular polygon. Round your answer to the nearest tenth.
a. An octagon with an apothem 4.8 cm long and a side 4 cm long.
b. A square with a side 24 in long and an apothem 12 in long.
Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
c. A pentagon with an apothem 316.6 mm long and a side 460 mm long.
d. A pentagon with a perimeter of 54.49m and an apothem of 7.5 m.
e. A hexagon with an apothem of 5√3 cm.
f. A square with an apothem of 10 2 cm.
Geometry
10.6 – Area of a Regular Polygon
Name_____________________
Date_____________________
For #2 – 5, find the area of the shaded region. Assume that polygons that appear to be
regular are regular.
2.
3.
8
6
5.5
8
4.
5.
2
4
12