Chapter 10Section 10.1 - Areas of

Parallelograms and Triangles

Objectives:

To find the area of a parallelogram

To find the area of a triangle

Theorem 10.1 – Area of a Rectangle The area of a rectangle is the product of its base and

height.

h

bA = bh

Base of a Parallelogram any of its sides Altitude a segment perpendicular to the line

containing that base, drawn from the side opposite the base.

Height the length of an altitude.

Base

Altitude

Theorem 10.2 – Area of a Parallelogram The area of a parallelogram is the product of a base

and the corresponding height.

h

b

A = bh

Base of a Triangle any of its sides

Height the length of the altitude to the line containing that base

Theorem 10.3 – Area of a Triangle The area of a triangle is half the product of a base and

the corresponding height.

h

b

A = bh

Ex: Find the area of each Parallelogram

4.5 in

4 in

5 in

4.6 cm 3.5

cm

2 cm

Ex: Find the area of each Triangle.

10 ft 4 ft

6.4 ft

13 cm5 cm

12 cm

When designing a building, you must be sure that the building can withstand hurricane-force winds, which have a velocity of 73 mi/h or more. The formula F = 0.004A gives the force F in pounds exerted by a wind blowing against a flat surface. A is the area of the surface in square feet, and v is the wind velocity in miles per hour. How much force is exerted by a 73 mi/h wind blowing against the side of the building shown below?

20 ft

12 ft

6 ft

Homework #18Due Tuesday (March 12)Page 536 – 538# 1 – 27 odd

Objectives:

To find the area of a trapezoidTo find the area of a rhombus or a kite

Section 10.2 – Areas of Trapezoids, Rhombuses, and Kites

Theorem 10.4 – Area of a Trapezoid The area of a trapezoid is half the product of the

height and the sum of the bases.

h

𝑏1

𝑏2

A = h()

Theorem 10.5 – Area of a Rhombus or a Kite The area of a rhombus or a kite is half the product of

the lengths of its diagonals.

𝑑1

𝑑2

A =

Ex: Find the area of the trapezoid

7 cm

12 cm

15 cm

What is the area of trapezoid PQRS? What would the area be if <P was changed to 45°?

7 m

5 m

60°P Q

RS

Ex: Find the area of the kite.

5 m

3 m

3 m

2 m

Find the area of the rhombus.

15 m

12 mA

B

C

D

E

Homework #19Due Wednesday (March 13)Page 542 – 543# 1 – 29 odd

Objectives:

To find the area of a regular polygon

Section 10.3 – Areas of Regular Polygons

You can circumscribe a circle about any regular polygon.

The center of a regular polygon is the center or the circumscribed circle.

The radius is the distance from the center to a vertex.

The apothem is the perpendicular distance from the center to a side.

Center

Radius

Apothem

Ex: Finding Angle Measures The figure below is a regular pentagon with radii and

apothem drawn. Find the measure of each numbered angle.

m<1 = = 72 (Divide 360 by the number of sides)m<2 = m<1 = 36 (apothem bisects the vertex angle)m<3 + 90 + 36 = 180 m<3 = 54

12

3

Find the measure of each angle of the half of an octagon.

12

3

Suppose you have a regular n-gon with side s. The radii divide the figure into n congruent isosceles triangles. Each isosceles triangle has area equal to (a being apothem/s being side).

Since there are n congruent triangles, the area of the n-gon is A = n · as. The perimeter p of the n-gon is ns. Substituting p for ns results in a formula for the area of the polygon.

Theorem 10.6 – Area of a Regular Polygon The area of a regular polygon is half the product of the

apothem and the perimeter.

pa

A = ap

Ex: Find the area of each regular polygon.

A regular decagon with a 12.3 in. apothem and 8 in. sides

A regular pentagon with 11.6 cm sides and an 8 cm apothem

Ex: Find the area of the hexagon.

10 mm

5 mm

Homework #20Due Thurs/Fri (March 14/15)

Page 548# 1 – 23 all

Objectives:

To find the perimeters and areas of similar figures

Section 10.4 – Perimeters and Areas of Similar Figures

Theorem 10.7 – Perimeters and Areas of Similar Figures

If the similarity ratio of two similar figures is , then1. the ratio of their perimeters is 2. the ratio of their areas is

Ex: Finding Ratios in Similar Figures The trapezoids below are similar. Find the ratio of their

perimeters and ratio of their areas.

6 m9 m

Ex: Two similar polygons have corresponding sides in the

ratio 5 : 7.

a. Find the ratio of their perimetersb. Find the ratio of their areas

Ex: Finding Areas Using Similar Figures The area of the smaller regular pentagon is 27.5 . What

is the area of the larger pentagon?

4 cm 10

cm

Ex:

The corresponding sides of two similar parallelograms are in the ratio 3 : 4. The area of the larger parallelogram is 96 . Find the area of the smaller parallelogram.

Ex: Finding Similarity and Perimeter Ratios

The areas of two similar triangles are 50 and 98 . What is the similarity ratio? What is the ratio of their perimeters?

The areas of two similar rectangles are 1875 and 135 . Find the ratio of their perimeters.

Homework #21Due Monday (March 18)Page 555 – 556# 1 – 23 all

Objectives:

To find the area of a regular polygon using trigonometry

To find the area of a triangle using trigonometry

Section 10.5 – Trigonometry and Area

In the last lesson, we learned how to find the area of

a regular polygon by using the formula A = ap. By using this formula and trigonometric ratios, you can solve other types of problems.

Trigonometry Review

Sine = SOH

Cosine = CAH

Tangent = TOA

Ex: Find the area of the regular pentagon with 8cm sides.

8 cm

Ex: Finding the Area and Perimeter

Find the area and perimeter of a regular octagon with radius 16m.

Suppose we want to find the area of triangle ABC (below), but you are only given m<A and lengths b and c. To use the formula A = bh, you need to find the height. This can be found by using the sine ratio:

sin A = therefore h = c(sin A)

A

B

Cb

hc a Area = bc(sin

A)

Theorem 10.8 – Area of a Triangle Given SAS The area of a triangle is one half the product of the

lengths of two sides and the sine of the included angle.

A

B

C

ca

b

Area ΔABC = bc(sin A)

Ex: Finding the area of a Triangle

Two sides of a triangular building plot are 120 ft and 85 ft long. They include an angle of 85°. Find the area of the building plot to the nearest square foot.

Homework #22Due Monday (March 18)Page 561 – 562# 1 – 27 odd

Quiz Tuesday (10.1 – 10.5)

Objectives:

To find the measures of central angles and arcs

To find circumference and arc length

Section 10.6 – Circles and Arcs

Circle the set of all points equidistant from a given point called the center.

Radius a segment that has one endpoint at the center and the other endpoint on the circle.

Congruent Circles have congruent radii

Diameter a segment that contains the center of a circle and has both endpoints on the circle

Central Angle an angle whose vertex is the center of the circle.

Circle = ΘP Central Angle = <CPA

Radius = CP Diameter = AB

A BP

C

Semicircle half of a circle (180°)

Minor Arc smaller than a semicircle (< 180°). Its measure is the measure of its corresponding angle.

Major Arc greater than a semicircle (> 180°). Its measure is 360 minus the measure of its related minor arc.

Adjacent Arcs arcs of the same circle that have exactly one point in common.

Ex: Identify the following in Θ O:

a. The minor arcs (4)b. The semicircles (4)c. The major arcs that contain point A

(4)

O

A

D

C

E

Postulate 10.1 – Arc Addition Postulate The measure of the arc formed by two adjacent arcs is

the sum of the measures of the two arcs.

A

B

C

mABC = mAB + mBC

Circumference the distance around a circle Pi (Π) the ratio or the circumference of a circle to its

diameter

Theorem 10.9 – Circumference of a Circle The circumference of a circle is Π (pi) times the diameter

dr

CO

C = Πd

C = 2Πr

Arc Length a fraction of a circle’s circumference

Theorem 10.10 Arc Length The length of an arc of a circle is the product of the

ratio and the circumference of the circle.

O

rA

B

Length of AB = · 2Πr

Ex: Find the measure of each arc.a. BC b. BD c. ABC d. AB

BC

D

A

O32°

58°

Ex:

The diameter of a bicycle wheel is 22in. To the nearest whole number, how many revolutions does the wheel make when the bicycle travels 100 feet?

Homework #23Due Thurs/Fri (March 21/22)Page 569 – 570# 1 – 39 odd

Objectives:

To find the areas of circles, sectors, and segments of circles.

Section 10.7 – Areas of Circles and Sectors

Theorem 10.11 – Area of a Circle The area of a circle is the product of Π (pi) and the

square of the radius.

r

O

A = Π

Sector of a Circle a region bounded by an arc of the circle and the two radii to the arc’s endpoints. A sector is named using one arc endpoint, the center of the circle, and the other arc endpoint.

Theorem 10-12 – Area of a Sector of a Circle The area of a sector of a circle is the product of the

ratio and the area of the circle.

A

O

r

B

Area Sector AOB = · Π

Segment of a circle a part of a circle bounded by an arc and the segment joining its endpoints. To find the area of a segment, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

T

Ex:

You’re hungry one day and decide to go to Warehouse Pizza to get some food. When you arrive, you check the menu and notice they are having deals on 14-in and 12-in pizzas. If a 14-in pizza costs $20.00 and a 12-in pizza costs $16.00, which price gives you the most pizza for your dollar?

Ex: Find the Area of a Segment of a Circle

A

BT

Find the area of the circle segment if the radius is 10-in and central angle ATB forms a right angle.

Objectives:

To use segment and area models to find the probabilities of events.

Section 10.8 – Geometric Probability

You may remember that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes

P(event) =

Geometric Probability a model in which points represent outcomes. We find probabilities by comparing measurements of sets of points.

P(event) =

Homework #24 Due Monday (March 25) Page 577 – 578

# 1 – 27 odd

Chapter 10 Test Tuesday/Wednesday

(Notebooks also due)