11.3 areas of regular polygons and 11.4 use geometric probability

11.3 Areas of Regular Polygonsand

11.4 Use Geometric Probability• You will find areas of

regular polygons inscribed in circles.

Essential Question:• How do you find the

area of a regular polygon?

You will learn how to answer this question by dividing the polygon into n isosceles triangles.

• You will use lengths and areas to find geometric probabilities.

•How do you find the probabilitythat a point randomly selected in aregion is in a particular part of thatregion?

You will learn how to answer this question by comparing the measure of the part of the region to the measure of the entire region.

Warm-Up ExercisesEXAMPLE 1 Find angle measures in a regular polygon

a. m AFB

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

AFB is a central angle, so m AFB = , or 72°.

360°

5


b. m AFG


SOLUTION

FG is an apothem, which makes it an altitude of isosceles ∆AFB. So, FG bisects AFB and m

AFG = m AFB = 36°.12


c. m GAF


SOLUTION

The sum of the measures of right ∆GAF is 180°.So, 90° + 36° + m GAF = 180°, and m GAF = 54°.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

In the diagram, WXYZ is a square inscribed in P.

1. Identify the center, a radius, an apothem, and a central

angle of the polygon.

P, PY or XP, PQ, XPY.

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 1

2. Find m XPY, m XPQ, and m PXQ.

90°, 45°, 45°

ANSWER

Warm-Up ExercisesEXAMPLE 2 Find the area of a regular polygon

DECORATING

You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15 inch sides and a radius of about 19.6 inches. What is the area you are covering?

SOLUTION

STEP 1 Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches.

Warm-Up ExercisesEXAMPLE 2

STEP 2

So, QS = (QP) = (15) = 7.5 inches.12

12

To find RS, use the Pythagorean Theorem for ∆ RQS.

a = RS ≈ √19.62 – 7.52 = 327.91 ≈ 18.108 √

Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP .

Find the area of a regular polygon


STEP 3 Find the area A of the table top.12A = aP Formula for area of regular polygon

≈ (18.108)(120)12 Substitute.

≈ 1086.5 Simplify.

Find the area of a regular polygon

So, the area you are covering with tiles is about 1086.5 square inches.

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

3.

Find the perimeter and the area of the regular polygon.

about 46.6 units, about 151.5 units2ANSWER

• You will find areas of regular polygons inscribed in circles.

Essential Question:• How do you find the

area of a regular polygon?• The center and radius of a

regular polygon are the center and radius of its circumscribed circle.• The distance from the center to a side of a regular polygon is theapothem.• A central angle of a regularpolygon is formed by twoconsecutive radii.• The area of a regular polygon isA =(1/2) where a is the apothemand P is the perimeter.

Find the length a of the apothem and the perimeter P of the polygon.Substitute those values into the formula for the area of a regular polygon

A =(1/2)aP

Warm-Up ExercisesEXAMPLE 1 Use lengths to find a geometric probability

SOLUTION

Find the probability that a point chosen at random on PQ is on RS .

0.6, or 60%.

Length of RS

Length of PQ

P(Point is on RS) =

=6

10,=

3

5

4 ( 2)

5 ( 5)

– –=

– –

Warm-Up ExercisesEXAMPLE 2 Use a segment to model a real-world probability

MONORAIL

A monorail runs every 12 minutes. The ride from the station near your home to the station near your work takes 9 minutes. One morning, you arrive at the station near your home at 8:46. You want to get to the station near your work by 8:58. What is the probability you will get there by 8:58?


SOLUTION

STEP 1

Find: the longest you can wait for the monorail and still get to the station near your work by 8:58. The ride takes 9 minutes, so you need to catch the monorail no later than 9 minutes before 8:58, or by 8:49. The longest you can wait is 3 minutes (8:49 – 8:46 = 3 min).


STEP 2

Model the situation. The monorail runs every 12 minutes, so it will arrive in 12 minutes or less. You need it to arrive within 3 minutes.

The monorail needs to arrive within the first 3 minutes.


STEP 3

Find: the probability.

P(you get to the station by 8:58)

Favorable waiting timeMaximum waiting time =

312 = =

14

The probability that you will get to the station by 8:58. is14

or 25%.

ANSWER


Find the probability that a point chosen at random on PQ is on the given segment. Express your answer as a fraction, a decimal, and a percent.

RT 1.

, 0.1, 10%ANSWER1

10


2. TS

12

, 0.5, 50%ANSWER



3. PT

25

, 0.4, 40%ANSWER



4. RQ

710

, 0.7, 70%ANSWER



5. WHAT IF? In Example 2, suppose you arrive at the station near your home at 8:43. What is the

probability that you will get to the station near your work by 8:58?

12

or 50%.ANSWER

Warm-Up ExercisesEXAMPLE 3 Use areas to find a geometric probability

The diameter of the target shown at the right is 80 centimeters. The diameter of the red circle on the target is 16 centimeters. An arrow is shot and hits the target. If the arrow is equally likely to land on any point on the target, what is the probability that it lands in the red circle?

ARCHERY

Warm-Up ExercisesEXAMPLE 3 Use areas to find a geometric probability

SOLUTION

Find the ratio of the area of the red circle to the area of the target.

P(arrow lands in red region) = Area of red circle

Area of target

=(82)(402)

=641600

=125

The probability that the arrow lands in the red region is125

, or 4%.

ANSWER

Warm-Up ExercisesEXAMPLE 4 Estimate area on a grid to find a probability

SCALE DRAWING

Your dog dropped a ball in a park. A scale drawing of the park is shown. If the ball is equally likely to be anywhere in the park, estimate the probability that it is in the field.

SOLUTION

STEP 1

Find the area of the field. The shape is a rectangle, so the area is bh = 10 3 = 30 square units.


STEP 2Find the total area of the park.

Make groups of partially covered squares so the combined area of each group is about 1 square unit. The total area of the partial squares is about 6 or 7 square units. So, use 52 + 6.5 = 58.5 square units for the total area.

Estimate area on a grid to find a probability

Count the squares that are fully covered. There are 30 squares in the field and 22 in the woods. So, there are 52 full squares.


STEP 3Write a ratio of the areas to find the probability.

Estimate area on a grid to find a probability

P(ball in field) = Area of field

Total area of park =300 585 =

2039

3058.5

The probability that the ball is in the field is about2039

, or 51.3%.

ANSWER


6. In the target in Example 3, each ring is 8 centimeters wide. Find the probability that an arrow

lands in a black region.

1425 = 56% ANSWER


7. In Example 4, estimate the probability that the ball is in the woods.

or about 48.7%.1939

ANSWER

• You will use lengths and areas to find geometric probabilities.

•How do you find the probabilitythat a point randomly selected in aregion is in a particular part of thatregion?• The probability of an event is a

measure of the likelihood that anevent will occur. It is a numberbetween 0 and 1, inclusive.• Geometric probability is a ratioinvolving lengths or areas.

Find the ratio of the measure of the specific part of the region to the measure of the entire region.

Warm-Up ExercisesDaily Homework Quiz

1. Find the measure of the central angle of a regular polygon with 24 sides.

Find the area of each regular polygon.

ANSWER 15°

2.

ANSWER 110 cm2


3.

ANSWER 374.1 cm2


Find the perimeter and area of each regular polygon.

4.

ANSWER 99.4 in. ; 745.6 in.2


5.

ANSWER 22.6 m; 32 m2


1. Find the probability that a point chosen at random on JN is on KN.

ANSWER

or 71.4%57


2. Find the probability that a randomly chosen point in the figure will lie in the shaded region.

ANSWER

36.3%


3. A bird is equally likely to be anywhere in thegarden shown. What is the probability that the bird will be in the vegetables?

ANSWER

about or 47%2451

11.3 areas of regular polygons and 11.4 use geometric probability

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