1. Measuring Length and area
Chapter 11
Measuring Length and area
11.1 areas of triangles and parallelograms areas of trapezoids, rhombuses, and kites perimeter and area of similar figures Circumference and arc length areas of circles and sectors areas of regular polygons use geometric probability

2. 11.1 Area of Triangles and Parallelogram
A = s2 s
Postulate 24: Area of Square Postulate
The area of the square is the square of the length of its sides.
Postulate 25: Area Congruence Postulate
If two polygons are congruent, then they have the same area.
Postulate 26: Area Addition Postulate
The area of the region is the sum of the areas of its non-overlapping parts.
Theorem 11.1: Area of the Rectangle
The area of the rectangle is the product of its base and height. h b
A = bh

3. 11.1 Area of Triangles and Parallelogram
Theorem 11.2: Area of a Parallelogram
The area of a parallelogram is the product of the base and
Its corresponding height. h b
A = bh
Theorem 11.3: Area of a Triangle
The area of a triangle is one half the product of the base and its corresponding height. b h
A = (1/2) bh
Parallelograms
Either pairs of the parallel sides can be used as the bases of the parallelogram. The height is the perpendicular distance between these bases. If you transform a rectangle to form other parallelograms with the same base and height, the area stays the same.

4. Chapter 11 Section 1 Practice Problems
Find the area of the shaded region.
Which postulate can be used to prove congruency? 4 8 12 4 A 4 B
Area Congruence Postulate
Area=(12)(8)-42
Area=96-16
Area=80 units2

5. Find the area of the shapes below.
Chapter 11 Section 1 Practice Problems continued…
Find the area of the shapes below. 5 9 3 6
A = bh
A = 9 ∙ 3 = 27
The area of the parallelogram is 27 units 2.
A = (1/2) bh
H = √(52 – 32) = 4
A = (1/2) 6 ∙ 4 = 12
The area of the triangle is 12 units2.

6. 11.2 Area of Trapezoids, Rhombuses, and Kites
The height of the trapezoid is the perpendicular distance between its bases. h b1 b2
A = (0.5)h(b1 + b2)
Theorem 11.4 Area of Trapezoid
The area of Trapezoid is one half the product
Of the height and the sum of the lengths of the bases.
A = (0.5)d1d2 d2
Theorem 11.5 Area of a Rhombus
The area of a Rhombus is one half the product of the
Length of its diagonals.
A = (0.5)d1d2 d2
Theorem 11.6 Area of a Kite
The area of a kite is the product of the lengths of its diagonals.

7. Chapter 11 Section 2 Practice Problems
The area of the trapezoid is 54ft2. Find the height of the trapezoid.
Find the area of the rhombus.
4ft
20ft
16ft
8ft
A = (0.5)h(b1 + b2)
54=(0.5)h(8+4)
54=(0.5)(12h)
54=(6h)
h=9ft
202=162+x2
X=12
d1=32ft, d2=24ft
A=(0.5)(32)(24)
A=384ft2

8. 11.3 Perimeter and Area of Similar Figures
Recall…
If two polygons are similar, then the ratio of their perimeters, or of any two corresponding lengths, is equal to the ratio of their corresponding side lengths.
However, the areas of similar polygons have a different ratio.
Theorem 11.7 Areas of Similar Polygons
The area of two polygons are similar with the lengths of corresponding sides in the ratio of a : b, then the ratio of their areas is a2 : b2.
Side length of Polygon I Side length of Polygon II
Area of Polygon I a2
Area of Polygon II b2 a b = I II a b
Polygon I ~ Polygon II
Any two regular polygons with the same number of sides are similar. Also, any two circles are similar.

9. Chapter 11 Section 3 Practice Problems
Find the height of figure B if it is similar to figure A (not drawn to scale obviously).
Write the ratios of the corresponding side lengths of the 2 similar polygons from figure A to figure B given their areas. A B
8ft
Area=40m2
Area of Figure A=108in2
Area of Figure B=432in2
= 1 4
Area=360m2
Ratio of Areas=1/9, so the ratios of corresponding side lengths in 1/3.
1/3=8/𝑥
X=24m.
The ratio of the corresponding sides in 1 2 , as the square root of is

10. 11.4 Circumference and Arc Length
The circumference of a circle is the distance around the circle. For all circles, the ratio of the circumference to the diameter is the same. This ratio is known as π, or pi. r d C
C = πd = 2πr
Theorem 11.8 Circumference of a Circle
The circumference C of a circle is C = πd or C = 2πr, where d is the diameter of the circle and r is the radius of the circle AB
Arc Length Corollary
In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360
X 2πr =
2πr
mAB
360
, or length of AB = B A

11. Chapter 11 Section 4 Practice Problems
Find the circumference of the circle. Express your answer in terms of π.
Find the length of AB. Express your answer to the nearest hundredth.
5in B A
30˚
12mm
Radius=5in, so diameter=10in
Circumference=πd
Circumference=10πin.
Circumference=24π
= 𝑥 24π
x=2π
x=6.28mm

12. 11.5 Area of Circles and Sectors
Theorem 11.9 Area of a Circle
The area of a circle is π times the square of the radius.
Sector of a circle: the region bounded by two radii of the circle and their intercepted arc.
Theorem Area of a Sector
The ratio of the area of the sector of a circle to the area of the whole circle (πr2) is equal to the ratio of the measure of the intercepted arc to 360◦ P B A r
x πr2
πr2
360 ◦ =
mAB
Area of sector APB
, or Area of sector APB =

13. Chapter 11 Section 5 Practice Problems
The area of sector ABC is 24πcm2. Find the measure of angle ABC.
Find the area of sector ABC to the nearest hundredth. A B C
45˚ 8m
12cm A C B
𝑥 64π =
X=8π
Area of sector ABC=25.13m2
Area of circle B=144πcm2
= 𝑥 360
m ∠ABC=60˚

14. 11.6 Areas of Regular PolygonsM Q
The center of the polygon (point P) and the radius of the polygon (MP) are the center and radius of the circle that circumscribes the regular polygon.
The distance from the center to any side of the polygon is called the apothem of the polygon (QP).
A central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices of the regular polygon (∠𝐌𝐏𝐍). s a =
so A 1 2
Theorem Area of a Regular Polygon
The area of a regular n-gon with side length s is one half the product of the apothem a and the perimeter P, aP
or A =
a * ns

15. Chapter 11 Section 6 Practice Problems
Find the area of the regular polygon. Express your answer to the nearest hundredth. (Hint=use trigonometric ratios to find the apothem.)
Find the area of the shaded region if the triangle is equilateral. Express your answer to the nearest hundredth.
6in
4ft
Apothem= 3 tan 36
15* 3 tan 36
Area=61.94in2
Area=29.48ft2

16. 11.7 Use Geometric Probability
The Probability of an event is the measure of the likelihood that the event will occur.
A geometric probability is a ratio that involves a geometric measure such as length or area.
Key concept Probability and Length
Let AB be a segment that contains the segment CD. If a point K on AB is chosen at random, then the probability that it is on CD is the ratio of the length of CD to the length of AB. B C D A
P(K is on CD) =
Length of CD
Length of AB
Key concept Probability and Area
Let J be a region that contains the region M. If a point K in J is chosen at random, then the probability that it is in region M is the ratio of the area of M to the area of J. M J
P(K is in region M) =
Area of M
Area of J

17. Chapter 11 Section 7 Practice Problems
What is the probability that a point randomly selected on AD is also on BC?
Express your answer as a common fraction.
What is the probability that a point randomly selected in region A is also in region B? Express your answer as a common fraction. (Region A is the entire figure). 13
182
A B C D A B
reduces to 1 14
The probability is 1 14
The probability is 1 4