#### Transcript Area of a Regular Polygon

Five-Minute Check (over Lesson 11–3)
CCSS
Then/Now
New Vocabulary
Example 1: Identify Segments and Angles in Regular Polygons
Example 2: Real-World Example: Area of a Regular Polygon
Key Concept: Area of a Regular Polygon
Example 3: Use the Formula for the Area of a Regular Polygon
Example 4: Find the Area of a Composite Figure by Adding
Example 5: Find the Area of a Composite Figure by Subtracting
Over Lesson 11–3
Find the area of the circle.
Round to the nearest tenth.
A. 37.7 ft2
B. 75.4 ft2
C. 223.6 ft2
D. 452.4 ft2
Over Lesson 11–3
Find the area of the sector.
Round to the nearest tenth.
A. 25.1 m2
B. 28.3 m2
C. 33.4 m2
D. 50.2 m2
Over Lesson 11–3
Find the area of the sector.
Round to the nearest tenth.
A. 506.8 in2
B. 570.2 in2
C. 760.3 in3
D. 1520.5 in2
Over Lesson 11–3
Find the area of the shaded region.
Assume that the polygon is regular.
Round to the nearest tenth.
A. 36.4 units2
B. 39.1 units2
C. 47.3 units2
D. 51.4 units2
Over Lesson 11–3
Find the area of the shaded region.
Assume that the polygon is regular.
Round to the nearest tenth.
A. 82.5 units2
B. 87.3 units2
C. 92.5 units2
D. 106.7 units2
Over Lesson 11–3
The area of a circle is 804.2 square centimeters.
The area of a sector of the circle is 268.1 square
centimeters. What is the measure of the central
angle that defines the sector?
A. 110°
B. 120°
C. 135°
D. 150°
Content Standards
G.MG.3 Apply geometric methods to solve
problems (e.g., designing an object or
structure to satisfy physical constraints or
minimize cost; working with typographic grid
systems based on ratios).
Mathematical Practices
1 Make sense of problems and persevere in
solving them.
6 Attend to precision.
You used inscribed and circumscribed figures
and found the areas of circles.
• Find areas of regular polygons.
• Find areas of composite figures.
• center of a regular polygon
• radius of a regular polygon
• apothem
• central angle of a regular polygon
• composite figure
Identify Segments and Angles in Regular
Polygons
In the figure, pentagon
PQRST is inscribed in
an apothem, and a central
angle of the polygon. Then
find the measure of a
central angle.
center: point X
apothem: XN
central angle: RXQ
Identify Segments and Angles in Regular
Polygons
A pentagon is a regular polygon with 5 sides. Thus,
the measure of each central angle of pentagon
PQRST is
or 72.
In the figure, hexagon ABCDEF is inscribed in
Find the measure of a central angle.
A. mDGH = 45°
B. mDGC = 60°
C. mCGD = 72°
D. mGHD = 90°
Area of a Regular Polygon
FURNITURE The top of the table
shown is a regular hexagon with
a side length of 3 feet and an
apothem of 1.7 feet. What is the
area of the tabletop to the
nearest tenth?
Step 1 Since the polygon has 6 sides, the polygon
can be divided into 6 congruent isosceles
triangles, each with a base of 3 ft and a
height of 1.7 ft.
Area of a Regular Polygon
Step 2
Find the area of one triangle.
Area of a triangle
b = 3 and h = 1.7
= 2.55 ft2
Simplify.
Step 3 Multiply the area of one triangle by the total
number of triangles.
Area of a Regular Polygon
Since there are 6 triangles, the area of the table is
2.55 ● 6 or 15.3 ft2.
UMBRELLA The top of an
umbrella shown is a regular
hexagon with a side length of
2 feet and an apothem of
1.5 feet. What is the area of
the entire umbrella to the
nearest tenth?
A. 6 ft2
B. 7 ft2
C. 8 ft2
D. 9 ft2
Use the Formula for the Area of a Regular
Polygon
A. Find the area of the
regular hexagon. Round to
the nearest tenth.
Step 1
Find the measure of a central angle.
A regular hexagon has 6 congruent central
angles, so
Use the Formula for the Area of a Regular
Polygon
Step 2
Find the apothem.
Apothem PS is the height of isosceles
ΔQPR. It bisects QPR, so mSPR = 30.
It also bisects QR, so SR = 2.5 meters.
ΔPSR is a 30°-60°-90° triangle with a
shorter leg that measures 2.5 meters, so
Use the Formula for the Area of a Regular
Polygon
Step 3
Use the apothem and side length to find the
area.
Area of a regular polygon
≈ 65.0 m2
Use a calculator.
Use the Formula for the Area of a Regular
Polygon
B. Find the area of the
regular pentagon. Round
to the nearest tenth.
Step 1
A regular pentagon has 5 congruent central
angles, so
Use the Formula for the Area of a Regular
Polygon
Step 2
Apothem CD is the height of isosceles
ΔBCA. It bisects BCA, so mBCD = 36.
Use trigonometric ratios to find the side
length and apothem of the polygon.
AB = 2DB or 2(9 sin 36°). So, the
pentagon’s perimeter is 5 ● 2(9 sin 36°).
The length of the apothem CD is 9 cos 36°.
Use the Formula for the Area of a Regular
Polygon
Step 3
Area of a regular polygon
a = 9 cos 36° and
P = 10(9 sin 36°)
Use a calculator.
A. Find the area of the regular hexagon. Round to
the nearest tenth.
A. 73.1 m2
B. 96.5 m2
C. 126.8 m2
D. 146.1 m2
B. Find the area of the regular pentagon. Round to
the nearest tenth.
A. 116.5 m2
B. 124.5 m2
C. 138.9 m2
D. 143.1 m2
Find the Area of a Composite Figure by Adding
POOL The dimensions of an irregularly shaped
pool are shown. What is the area of the surface of
the pool?
The figure can be separated into a rectangle with
dimensions 16 feet by 32 feet, a triangle with a base of
32 feet and a height of 15 feet, and two
semicircles with radii of 8 feet.
Find the Area of a Composite Figure by Adding
Area of composite figure
 953.1
Answer: The area of the composite figure is
953.1 square feet to the nearest tenth.
Find the area of the figure
in square feet. Round to the
nearest tenth if necessary.
A. 478.5 ft2
B. 311.2 ft2
C. 351.2 ft2
D. 438.5 ft2
Find the Area of a Composite Figure by
Subtracting
Find the area of the shaded figure.
To find the area of the figure, subtract the area of the
smaller rectangle from the area of the larger rectangle.
The length of the larger rectangle is 25 + 100 + 25 or
150 feet. The width of the larger rectangle is 25 + 20 +
25 or 70 feet.
Find the Area of a Composite Figure by
Subtracting
= area of larger rectangle – area of smaller rectangle
Area formulas
Substitution
Simplify.
Simplify.