GT/Honors Geometry
March 20 , 2016 – April 5. Areas of 2-D figures
Topic
Review – Proving Statements on Circles
WS
Date
Monday
03/20
Tuesday
Herons formula and area of Triangles
03/21
Wednesday 13.4 Dimensional Change: Review area of polygons and investigate
03/22
relationships of dimensional change.
Thursday
03/23
13.4 and 13.1 Review area of polygons and investigate relationships of
dimensional change.
Friday
03/24
Apply dimension change to solve problems
Monday
03/27
Tuesday
03/28
Assignment
ws
pp. 543-544: 1-15
Pps. 544-545: 16-24(even), 2829.
p. 545: 25 - 33
Quiz- Dimension change and Heron’s Formula
Composite Area: (G.9A, G 11A, G.11B) Derive and Apply Area of
Regular and other polygons, Determining area of composite figures
(Parallelograms, Trapezoids, and Triangles)
Determining area of composite figures Day 2
Wednesday 13.3 Areas of regular polygons using apothem and trig.
03/29
Thursday Solve for area using the apothem
03/30
Friday.
Solve for area using the apothem
03/31
Solve for area using the apothem
Pg. 548-549: 9-14, 22, WS 1:1-4
Complete WS 2
Pg. 536: 1-12, p. 548: 18-21
WS
Ws: Areas and composite figures.
Pg. 549: 22-34
Areas of composite figures.
Quiz- on Area of Regular polygons
Monday
03/28
Tuesday
03/29
Wednesday
03/30
Thursday
3/31
Friday
04/01
15.2. Probability and Linear probability ratios. Length Probability Ratio to
show probability in a geometric concept.
15.2 Area probability ratios of geometric probability. Quiz 2: 13.4 to 13.5,
15.2
Test Review: 13.1 – 13.5, 15.2
14.1 Cross sections: Students will determine what 3D figure was sliced
given a cross section, Euler’s formula: F + V = E + 2
Pg. 563: 1-17 and Ws
Monday
04/04
14.1 Students will identify which 3D object is formed when rotating a 2D
shape
Ws and pg. 564: 18-32
Test areas of 2-D Objects and probability.
Pg. 619: 1-34 (odd)
Pg. 619: 1-34 (Even)
Pg. 551: 1-33
Read. Pg. 558-559.
1
Study Guide 13-1 Areas of Parallelograms and Triangles
Area of a Rectangle is the product of its base and height.
A = bh
Area of a Parallelogram is the product of a base and the corresponding height.
Height
A = bh
Base
The base of a parallelogram is any of its sides. The corresponding
altitude is a segment perpendicular to the line containing that base,
drawn from the side opposite the base. The height is the length of
an altitude.
Height
Base
Example 1: Find the area of the parallelogram.
8m
Example 2: Find the value of h.
8
10.5 m
7
10
12 m
Example 3: Find the value of h.
h
h
4
5
Example 4: A parallelogram has 9-in. and 18 in.
sides. The height corresponding to the 9-in.
base is 15 in. Find the height corresponding to
the 18-in. base.
4.3
Example 5: What is the area of ABDC with vertices
A(-4, -6), B(6, -6), C(-1, 5), and D(9, 5)
2
Area of a Triangle is half the product of a base and the corresponding height.
A=
1
bh
2
h
A base of a triangle is any of its sides. The corresponding height is
the length of the altitude to the line containing that base.
b
b
h
Example 6: Find the area of the triangle.
13 cm
Example 7: Find the area of the triangle.
31 cm
7 ft
4 ft
30 cm
Example 8: Find the area of the shaded region.
2 ft
Example 9: Find the area of the shaded region.
22
3
14
7
10
16
Example 10: Graph the lines and find the area of the triangle
enclosed by the lines.
y=7
x = -3
y = 2x – 1
Example 11: Find the area of an isosceles triangle whose legs are 13 cm each and whose base is 10
cm.
3
Notes – Use Heron’s Formula to find the area of a triangle
If you know the lengths of the sides of a triangle, you can use Heron’s formula to find the area.
The area, A, of a triangle with sides measuring a, b, and c is:
A=
s (s  a )(s  b )(s  c ) , where s =
a b c
2
Example
1. Use Heron’s formula to find the area of the triangle.
(sometimes called the semi-perimeter)
9
5
12
2. Use Heron’s formula to find the area of a triangle with sides 4, 6, and 8.
3. Find x:
9
x
17
10
4
WS 1.Geometry Worksheet
Heron’s Formula
1. Given a triangle with sides 10, 12, and 14
inches long, find the length of the altitude
upon the 12 inch side.
Name___________________________________
Date_______________________Period________
2. Find the area of a parallelogram with two adjacent
sides 8 inches and 10 inches long and one diagonal
12 inches long.
3. Find the area of quadrilateral RSTV if mR =
90, RS = 12, ST = 8, TV = 7, and VR = 5
4. Given a triangle with sides 4, 6 and 8 inches long,
find the length of the altitude upon the shortest side.
5.
6.
8.
7.
5
Geometry
Name_____________________________________
WS 2: Worksheet
Date_______________________Period__________
1. The area of a trapezoid is 80 square units. If
2. A rhombus has an angle measure of 120, and its
its height is 8 units, find the length of its
shorter diagonal has a length of 10 inches.
median.
Find the area of the rhombus.
3.
A developer wanted to blacktop the shaded portion of this corner building site. How much will it cost
if blacktopping costs $5 per square meter?
48 m
16 m
14 m
8m
4.
The area of an isosceles trapezoid is 76 square
inches, the height is 4 inches, and the
congruent sides of the trapezoid are 5 inches
long. Find the lengths of the bases.
5.
Suppose the perimeter of rhombus ABCD is 20
and AC = 8. Find the area of the rhombus.
6.
A city planner measured the distances shown from the old oak, the big rock, and the flag to the eastwest line. Other distances along the line are shown. What is the area of the city park?
Big
Rock
Flag
Old
Oak
City Park
164 m
126 m
83 m
W
7.
62 m
48 m
106 m
E
The area of an 8 by 10 rectangular picture and
its white border of uniform width is 120 u2.
How wide is the border?
8.
The shorter diagonal of a rhombus has the
same length as a side. Find the area if the
longer diagonal is 12 cm. long.
6
9.
Find the area of an isosceles triangle with base
angles each measuring 30° and the legs
measuring 10 cm. each.
10. Find the area of trapezoid WXYZ.
18
X
Y
13
12
Z
W
11. Find the total area of the parallelogram.
12. Find the area of the shaded region.
17
8
15
17
8
10
10
17
17
13. A walk 3 feet wide surrounds a rectangular
grass plot 30 feet long and 18 feet wide. Find
the area of the walk.
14. Find the area of the shaded region.
17
8
5
15. Find the area of the parallelogram.
10
16. Find the area of the trapezoid.
6
8
8 2
60°
11
45°
7
Notes for sections 13.3 – Area of Regular Polygons
In a regular polygon, a segment drawn from the center of the
polygon perpendicular to a side of the polygon is called an
apothem. In the figure at the right PS is an apothem.
A segment drawn from the center of the polygon to a vertex is
called a radius of the polygon.
In the figure at the right, PA is a radius.
A
P
S
he area of a regular polygon can be found by dividing the polygon
into congruent isosceles triangles. For example, the pentagon above
can be divided into 5 triangles by drawing all five radii. If a regular
polygon has n sides then it can be divided into n triangles.
Now find the area of one of the triangles (note: area of triangle = ½ bh). The height of the triangle will be the
apothem. The base of the triangle is the length of one side, s, of the polygon.
Therefore, area of the triangle = ½ sa.
Since there are n triangles, multiply the area of the triangle by n to get the area of the polygon.
Area of polygon = n[1/2sa] or ½ nsa
However, ns is just the perimeter, P, of the polygon. Therefore, the area of a regular polygon with perimeter P
and apothem a is
A = ½ Pa
Ex. 1 Find the area of a regular pentagon with
Ex. 2 Find the area of a regular hexagon with an
perimeter 54.49 m and an apothem of 7.5 m.
apothem of 5 3 cm and each side 10 cm.
Equilateral triangles are sometimes easier to work with A 
s2 3
4
Ex. 3 Find the apothem, area, and perimeter of the equilateral triangle below.
7
8
Ex. 4 Hexagons are made up of 6 congruent equilateral triangles so the area of a regular hexagon can be
 s2 3 
 .
 4 
found using: A  6  
Find the apothem, perimeter, and area of a regular hexagon that has a side
of 8 in.
Find the apothem, perimeter, and area of each regular polygon. Round your answers to the nearest tenth.
Ex. 5
Ex. 6 radius = 5 in.
5 in
10 m
9
Note – 13-4 Perimeters and Areas of Similar Figures
If the similarity ratio of two similar figures is
(1) the ratio of their perimeters is
(2) the ratio of their areas is
a2
.
b2
a
, then
b
a
and
b
4
Example 1: The triangles at the right are similar.
(a) Find the ratio (larger to smaller) of the perimeters.
5
(b) If the perimeter of the smaller triangle is 18 cm, find the perimeter of the larger triangle.
(c) Find the ratio (larger to smaller) of the areas.
(d) If the area of the larger triangle is 410 cm2, find the area of the smaller triangle.
Example 2: The ratio of the lengths of the corresponding sides of two regular octagons is
of the larger octagon is 320 ft2. Find the area of the smaller octagon.
8
. The area
3
Example 3: Benita plants the same crop in two rectangular fields, each with side lengths in a ratio of
1
2:3. Each dimension of the larger field is 3 times the dimension of the smaller field. Seeding
2
the smaller field costs $8. How much money does seeding the larger field cost?
Example 4: The areas of two similar polygons are 32 in.2 and 72 in.2 If the perimeter of the smaller
polygon is 15 in, find the perimeter of the larger polygon.
10
GT/Honors Geometry
Chapter 13 Area and composite figures
Name_______________________________
Date___________________Period________
For the following problems, find the area of the entire figure if nothing is shaded, or find the area of the shaded
region if there is one. All answers should be exact unless you are asked to round.
1. A=_____________
2. A=___________ The area of a circle is 24π
cm2. Find the circumference of this circle.
15
10
45˚
3. A=________________
4. A=______________ Equilateral Triangle
14
9
17
5. A=________________; p = ______________
Regular hexagon with semicircles attached
4 3
6. A=_________
A
B
E
8
D
7. A=__________ Find the area of a regular nonagon
with sides of length 12 cm. Round to the nearest
tenth.
ABCD is a rhombus with AC=18
41
C
8. A=__________
Regular Polygon, round to the nearest tenth.
8
11
9. A=____________
10. Given: mABC=100˚, CB = 15; A=__________;
arc length = __________
24
B
A
12 11
C
11. A=_________ P = _______ probability of
landing in the shaded area (nearest hundredth) =
__________
__________
60˚
6
12. A=___________ ΔABC is equilateral.
A
BD=DC=10. mDCB = 30˚.
C
B
D
13. Find the area of an isosceles trapezoid that has
bases 8 cm. and 18 cm. and that has legs that are 13
cm. long.
14. Find the area of MNP if m = 14, p = 16, and
mMNP = 63. Round your answer to the
nearest tenth.
15. Find the area of the triangle bounded by
16. Find the area of the quadrilateral ABCD with
coordinates A(-1,5), B(6,4), C(2,-2), D(-5,2).
x = -2, y = 4 and y = 3x – 2.
A=__________
A=__________
17.
B
15
A
AF=FG; FC=ED=FE=CD.
9
C
F
G
E
D
semicircle
A=______________
P = _____________
12
18. Joey has his dog on a leash tied to the corner of the garage. The garage is 20 feet by 30 feet. If
the leash is 24 feet long, how much area can the dog access?
30
20
24
Find the area of the shaded portion in each figure. The “dots” are centers of circles.
19.
21.
20.
22.
13
14
Notes – 15.2 – Geometric Probability
Length Probability Postulate:
If a point on AB is chosen at random and C is between A and B, then the probability that the point is on
AC is:
length AC
length AB
Ex.
D
E
0
1
F
2
3
4
5
G
H
6
7
I
8
9
What is the probability a point chosen at random on DI is also a part of:
(a) EF
(b) FI
Area Probability Postulate
If a point in region A is chosen at random, then the probability that the point is in region B, which is in
the interior of region A, is:
area of region B
area of region A
Ex. Joanna designed a new dart game. A dart in section A earns 10 points; a dart in section B earns 5
points; a dart in section C earns 2 points. Find the probability of earning each score.
radius of circle A = 2 in.
radius of circle B = 5 in.
C
radius of circle C = 10 in.
B
A
15
Ex. Find the probability that a point chosen at random in this circle will be in the given section.
(a) A
E
(b) C
D
(c) D
C
A
50°
60° 120°
B
Ex. Find the probability that a point chosen at random in each figure lies in the shaded region. Round
your answer to the nearest hundredth.
(a)
(b)
8 cm
Regular hexagon (sides = 12 cm)
inside a rectangle
16
GT/Honors Geometry
Chapter 13 Review – Area
Name_______________________________
Date___________________Period________
For the following problems, find the area of the entire figure if nothing is shaded, or find the area of the shaded
region if there is one. All answers should be exact unless you are asked to round.
1. A=_____________
2. A=___________ The area of a circle is 24π
cm2. Find the circumference of this circle.
15
10
45˚
3. A=________________
4. A=______________ Equilateral Triangle
14
9
17
5. A=________________; p = ______________
Regular hexagon with semicircles attached
4 3
6. A=_________
A
B
E
8
D
7. A=__________ Find the area of a regular nonagon
with sides of length 12 cm. Round to the nearest
tenth.
ABCD is a rhombus with AC=18
41
C
8. A=__________
Regular Polygon, round to the nearest tenth.
8
17
9. A=____________
10. Given: mABC=100˚, CB = 15; A=__________;
arc length = __________; probability of landing
in the shaded area (exact) = __________
24
B
A
12 11
C
11. A=_________ P = _______ probability of
landing in the shaded area (nearest hundredth) =
__________
__________
60˚
6
12. A=___________ ΔABC is equilateral.
A
BD=DC=10. mDCB = 30˚.
C
B
D
13. Find the area of an isosceles trapezoid that has
bases 8 cm. and 18 cm. and that has legs that are 13
cm. long.
14. Find the area of MNP if m = 14, p = 16, and
mMNP = 63. Round your answer to the
nearest tenth.
15. Find the area of the triangle bounded by
16. Find the area of the quadrilateral ABCD with
coordinates A(-1,5), B(6,4), C(2,-2), D(-5,2).
x = -2, y = 4 and y = 3x – 2.
A=__________
A=__________
17.
B
15
A
AF=FG; FC=ED=FE=CD.
9
C
F
G
E
D
semicircle
A=______________
P = _____________
18
18. Joey has his dog on a leash tied to the corner of the garage. The garage is 20 feet by 30 feet. If
the leash is 24 feet long, how much area can the dog access?
30
20
24
Answers: (1) 75 2 u2 ; (2) 4 6  cm ; (3) 6 110 u2 ; (4) 144 3 u2 ; (5) (96 3  24)u2 ; (24 + 12  )u; (6) 720
u2; (7) 890.2 cm2; (8) 110.1 u2 ; (9) 36 55 u2 ; (10)
125 2 25
 u ; 5/18 (11) (6  9 3)u2 ;
u ;
3
2
(6 + 2  ) u; .03; (12) 100 3 u2 ; (13) 156 cm2; (14) 99.8 cm2 ; (15) 24 u2; (16) 41.5 u2;
(17) (351  28.125)u2 , (63 + 7.5  ) u; (18) 436  ft2 (Review quizzes)
19
Cross Sections of Solids
Cross sections of a solid occur when a plane intersects the solid. Cross sections are typically formed by slicing
parallel to a face, slicing through an axis of symmetry, across a corner, etc.
The cube or “tall cube” is filled with colored water to simulate the cross sections. It has been filled to a level to
allow you to see the more difficult cross sections. (There may be some that you cannot see with the colored water
because there is too much water.) Determine which, if any, of the shapes listed below can be cross sections.
Sketch the cross section on the cube provided.
CAUTION: The solids may leak if you tip them so that the top seam has water on it.
Square
Rectangle
Trapezoid
Isosceles Trapezoid
Pentagon
Hexagon
Scalene Triangle
Isosceles Triangle
that is not equilateral
Equilateral Triangle
Parallelogram
Rhombus
20
14-1 Space Figures and Cross Sections
Polyhedron
* 3D figure whose surfaces are _______________________
* each polygon is a _______________.
* an ____________ is a segment where two faces intersect.
* a _____________ is a point where three or more edges intersect.
Example 1: List the vertices, edges, and faces of the polyhedron.
A
B
F
E
C
D
Leonhard Euler (1707-1783), a Swiss mathematician, discovered a relationship among the numbers of
faces, vertices, and edges of any polyhedron.
Euler’s Formula:
F + V = E + 2 where F = # of faces
V = # of vertices
E = # of edges
Example 2: Find the number of edges of a polyhedron with 6 faces and 8 vertices.
Note: In two-dimensions, Euler’s formula reduces to
F+V=E+1
where F is the number of regions formed by V vertices linked by E segments.
E
Consider a net for the polyhedron in Example 1.
Regions = _________
A
D
E
A
B
C
F
B
Vertices = _________
Segments = __________
F
21
A cross section is the intersection of a solid and a plane. You can think of a cross section as a very thin
slice of a solid.
Describe each cross section:
22
Name__________________________________
Date____________________Period__________
Solids Formed by Rotation of 2-D
Graph of Region
Describe and draw the solid formed
when the figure is rotated about
y-axis
Describe and draw the solid formed
when the figure is rotated about
x-axis
y
y
x
y
x
y
x
y
x
y
x
y
x
y
x
x
23
Solids Formed by Rotation
Graph of Region
Describe and draw the solid formed
when the figure is rotated about
y-axis
Describe and draw the solid formed
when the figure is rotated about
x-axis
y
y
x
y
x
y
x
y
x
y
x
y
x
y
x
x
24