Center: point F, radius:
, apothem:
, central
angle:
, A square is a regular polygon with 4
sides. Thus, the measure of each central angle of
11-4 Areas of Regular Polygons and Composite Figures
square ABCD is
or 90.
Find the area of each regular polygon. Round to
the nearest tenth.
1. In the figure, square ABDC is inscribed in F.
Identify the center, a radius, an apothem, and a
central angle of the polygon. Then find the measure
of a central angle.
2. SOLUTION: An equilateral triangle has three congruent sides.
Draw an altitude and use the Pythagorean Theorem
to find the height.
SOLUTION: Center: point F, radius:
, apothem:
, central
angle:
, A square is a regular polygon with 4
sides. Thus, the measure of each central angle of
square ABCD is
or 90.
Find the area of each regular polygon. Round to
the nearest tenth.
2. SOLUTION: An equilateral triangle has three congruent sides.
Draw an altitude and use the Pythagorean Theorem
to find the height.
Find the area of the triangle.
3. SOLUTION: The polygon is a square. Form a right triangle.
Find the area of the triangle.
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11-4 Areas of Regular Polygons and Composite Figures
4. POOLS Kenton’s job is to cover the community pool
during fall and winter. Since the pool is in the shape
of an octagon, he needs to find the area in order to
have a custom cover made. If the pool has the
dimensions shown at the right, what is the area of the
pool?
3. SOLUTION: The polygon is a square. Form a right triangle.
SOLUTION: Since the polygon has 8 sides, the polygon can be
divided into 8 congruent isosceles triangles, each with
a base of 5 ft and a height of 6 ft.
Find the area of one triangle.
Use the Pythagorean Theorem to find x.
Since there are 8 triangles, the area of the pool is 15
· 8 or
120 square feet.
Find the area of the square.
CCSS SENSE-MAKING Find the area of each
figure. Round to the nearest tenth if necessary.
4. POOLS Kenton’s job is to cover the community pool
during fall and winter. Since the pool is in the shape
of an octagon, he needs to find the area in order to
have a custom cover made. If the pool has the
dimensions shown at the right, what is the area of the
pool?
SOLUTION: Since the polygon has 8 sides, the polygon can be
divided into 8 congruent isosceles triangles, each with
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a base
of 5- ft
and a by
height
of 6 ft.
Find the area of one triangle.
5. SOLUTION: Page 2
11-4 Areas of Regular Polygons and Composite Figures
2
Therefore, the area of the figure is about 71.8 in .
In each figure, a regular polygon is inscribed in
a circle. Identify the center, a radius, an
apothem, and a central angle of each polygon.
Then find the measure of a central angle.
6. SOLUTION: To find the area of the figure, subtract the area of th
triangle from the area of the rectangle.
8. SOLUTION: Center: point X, radius:
, apothem:
, central
angle:
, A square is a regular polygon with 4
sides. Thus, the measure of each central angle of
Use the Pythagorean Theorem to find the height h of
isosceles triangle at the top of the figure.
square RSTVW is
or 72.
9. SOLUTION: Center: point R, radius:
, apothem:
, central
angle:
, A square is a regular polygon with 4
sides. Thus, the measure of each central angle of
Area of the figure = Area of rectangle – Area of tria
square JKLMNP is
or 60.
Find the area of each regular polygon. Round to
the nearest tenth.
2
Therefore, the area of the figure is about 71.8 in .
In each figure, a regular polygon is inscribed in
a circle. Identify the center, a radius, an
apothem, and a central angle of each polygon.
Then find the measure of a central angle.
10. SOLUTION: An equilateral triangle has three congruent sides.
Draw an altitude and use the Pythagorean Theorem
to find the height.
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Center: point X, radius:
, apothem:
, central
angle:
, A square is a regular polygon with 4
Page 3
Center: point R, radius:
, apothem:
, central
angle:
, A square is a regular polygon with 4
sides. Thus, the measure of each central angle of
11-4square
AreasJKLMNP
of Regular
is Polygons
or 60.and Composite Figures
11. SOLUTION: The formula for the area of a regular polygon is
, so we need to determine the perimeter
and the length of the apothem of the figure.
A regular pentagon has 5 congruent central angles,
Find the area of each regular polygon. Round to
the nearest tenth.
so the measure of central angle is
or 72.
10. SOLUTION: An equilateral triangle has three congruent sides.
Draw an altitude and use the Pythagorean Theorem
to find the height.
Apothem
is the height of the isosceles triangle ABC. Triangles ACD and BCD are congruent, with
∠ACD = ∠BCD = 36.
Use the Trigonometric ratios to find the side length
and apothem of the polygon.
Find the area of the triangle.
Use the formula for the area of a regular polygon.
11. SOLUTION: The formula for the area of a regular polygon is
, so we need to determine the perimeter
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and the length
of thebyapothem
A regular pentagon has 5 congruent central angles,
so the measure of central angle is
or 72.
12. SOLUTION: Page 4
A regular hexagon has 6 congruent central angles
that are a part of 6 congruent triangles, so the
11-4 Areas of Regular Polygons and Composite Figures
13. 12. SOLUTION: A regular octagon has 8 congruent central angles,
from 8 congruent triangles, so the measure of central
angle is 360 ÷ 8 = 45.
SOLUTION: A regular hexagon has 6 congruent central angles
that are a part of 6 congruent triangles, so the
measure of the central angle is
= 60.
Apothem
is the height of the isosceles triangle ABC and it splits the triangle into two congruent
triangles. Apothem
is the height of equilateral triangle
ABC and it splits the triangle into two 30-60-90
triangles.
Use the Trigonometric ratio to find the side length of
the polygon.
Use the trigonometric ratio to find the apothem of the
polygon.
AB = 2(AD), so AB = 8 tan 30.
CCSS SENSE-MAKING Find the area of each
figure. Round to the nearest tenth if necessary.
13. SOLUTION: A regular octagon has 8 congruent central angles,
from 8 congruent triangles, so the measure of central
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angle is 360 ÷ 8 = 45.
15. SOLUTION: Page 5
11-4 Areas of Regular Polygons and Composite Figures
CCSS SENSE-MAKING Find the area of each
figure. Round to the nearest tenth if necessary.
17. SOLUTION: 15. SOLUTION: 16. SOLUTION: eSolutions Manual - Powered by Cognero
17. Page 6