1
Area Formulas of Triangles and Quadrilaterals Day 1 Classwork
Definitions:
Area is what covers the figure.
Altitude is the height of a figure.
An altitude is needed when a figure doesn’t have a side with a 90 0 angle.
Formulas:
Triangle h h A = 2
1bh or A =
2
bh
b b
Rectangle h A = bh or A = lw
b
Square A = bh or A = s 2
s
Parallelogram h A = bh
b
b1
Trapezoid h A = 2
1(b 1 + b 2 ) h or A =
2
)( 21 hbb
b 2
**When finding the area of a figure, always square your units!
2
Area Formulas of Triangles and Quadrilaterals Day 1 Classwork
Find the Area
1) 2) 3)
8m 7m 6m
12m 15m 9m
4) 3cm 5) 6m
4cm 7m
7cm
6) 7) 8)
7.8m 4.1m 6m
12.25m 15.3m 3.5m
Find the missing measures for each rectangle.
9) l = 14 in. 10) l = 11) l = 5m
w = w = 2cm w =
A = A = A = 60 m2
P = 34 in P = 25cm P =
12) Find the area of the triangle 13) Find the area of quadrilateral 𝑨𝑩𝑪𝑫 two different
3
Area Formulas of Triangles and Quadrilaterals Day 1 Classwork
Find the area for the following:
1) 2) 3) 4)
9m 6m 3.25m 4.5ft
12m 10m 8.4m 3ft
5) A rectangular fish pond is 21ft2
in area. If the owner can surround the pond with a 20 foot fence,
what are the dimensions of the pond?
Multiple Choice Questions
6) A rectangular playground is 85 feet long and 60 feet wide. What is the area of the playground?
A. 290 ft2
B. 2, 550 ft2
C. 510 ft2
D. 5,100 ft2
7) Which of the figures below have the same area?
5m
8m 4m 4m 6m
7m 6m 12m 9m
Figure 1 Figure 2 Figure 3 Figure 4
A. Figures 1 and 2 B. Figures 2 and 3 C. Figures 2 and 4 D. Figures 3 and 4
8) Which figure has the least area?
A. Square with a side length of 9 cm
B. Parallelogram with a base of 12cm and height of 6cm
C. Triangle with a base of 18cm and a height of 6cm
D. Rectangle with a width of 7cm and a length of 8cm
4cm
9) Aster made a sticker in the shape shown to the right. 1.2cm
What is the area of the sticker?
3cm
1.5cm
4
Similar Figures Day 2 Classwork
Similar figures have the same shape, but NOT necessarily the same size. For this
reason, all congruent figures are similar, but not all similar figures are congruent.
Similar Figures have the following properties:
Their Corresponding sides have proportional lengths.
Their corresponding angles are congruent.
Corresponding means -
_______________________________________________________________________
_______________________________________________________________________
Example:
*Angles are congruent since all
5cm 2.5cm rectangles have four right angles.
7.5cm 3.75cm
The ratios of the lengths and widths of the rectangles are in proportion.
75.3
5.7 =
5.2
5 (Cross Multiply) *Therefore, the rectangles are similar because,
18.75 = 18.75 the rectangles have congruent angles and
corresponding sides are in proportion.
State whether the following are Similar Figures or Not.
1) 2) 8in
6m
1.5m 12in 16in
4m 1m 3in
5
Similar Figures Day 2 Classwork
The following figures are similar. Find the missing side. 40in
1. 2. 3. 8m 4m 9cm 6cm 96in 30in
10m x 4cm x
x
Multiple Choice Questions
4. A gate is 3 feet and casts a shadow 5 feet long. At the same time, a nearby building casts a shadow
45 feet long. What is the height of the building?
A. 15 feet B. 27 feet C. 43 feet D. 75 feet
5. Which describes these two figures?
A. Congruent but not similar
B. Neither similar nor congruent
4m 3m C. Similar but not congruent
D. Similar and congruent
6m 4.5m
6. The following right triangles are similar. Find the length for the side representing x.
A. 72 cm
B. 78 cm
C. 128 cm
96 cm 104cm x D. 139 cm
40cm 30cm
6
Similar Figures Day 2 Classwork
Prove if the following polygons are similar or not.
1) 2)
12.5m
3.1m 8m 20m
7.5m 2m 3m 7.5m
Each pair of figures below is similar. Find the value of each variable.
3) 4) 12m n
30m 54m 6m 3m
45m y
5) A woman is 5 ft. tall and her shadow is 4 ft. long. A nearby tree has a shadow 30 ft. long. How tall
is the tree?
6) Paula casts a shadow 2 meters long at the same time a tree casts a shadow 28 meters long. The tree
is 17.5 meters tall. How tall is Paula?
Review
Solve and Check
7) 5x – 9 + 10 = 31 8) -2( 4x + 9) = 30 9) 8x – 10x + 8 = -20
7
Similar Figures with Perimeter and Area Day 3 Classwork
Review- State the formula for each of the following.
Perimeter of any figure: _______________________
Area of a Triangle: ___________________________
Area of a Parallelogram: _______________________
Area of a Trapezoid: __________________________
Steps to finding the Perimeter or Area of Similar Figures:
Always find the missing side first. (Set up a proportion to find missing side)
When all sides are present, use the correct formula to find what is asked.
Find the perimeter of each of the following similar figures.
1) 2)
14m 14m
7m 7m 1.5m x
10m x 2m 4m
Find the area of each of the following similar figures.
3) 4)
15m 2m 3cm 15cm
30m x 4cm x
8
Similar Figures with Perimeter and Area Day 3 Classwork
5) Justin has two rectangular photo prints that are similar. The length of the smaller
print is 5 inches and the width is 3 inches. The length of the larger print is 20 inches.
a) Find the width of the larger print.
b) Find the perimeter of the smaller print.
c) Find the Area of the larger print.
6) The ratio of the corresponding sides of two similar triangles is 4 : 9. The sides of the
smaller triangle are 10cm, 16cm, and 18cm.
a) Find the three dimensions of the larger triangle.
b) Find the perimeter of the larger triangle.
7) Surveyors know that the two triangles to the right are similar. They cannot measure the distance d
across the lake directly. Find the distance across the lake.
2.32m
2.16m
Lake
(d) 1.74m
9
Similar Figures with Perimeter and Area Day 3 Classwork
ABC is similar to PQR. Find each measure.
C R 1) Length of AB
21 ft 14 ft 2) Length of RP
16.8 ft 3) Measure of A
530 B 92
0 Q 4) Measure of Q
P 8 ft
A
5) An image is 16 in. by 20 in. You want to make a copy that is similar. Its longer side will be 38 in.
The copy costs $0.60 per square inch. Estimate the copy’s total cost.
6) You want to enlarge a copy of the flag of the Philippines that is 4 in. by 8 in. The two flags will be
similar. How long should you make the shorter side if the long side is 24 ft.?
Find the Perimeter of each figure. Find the Area of each figure
7) 8)
x 4cm 6m 9m
8m x 12.5cm 5cm
9) Give three examples of real-world objects that are similar. Explain why they are similar.
___________________________________________________________________________________
___________________________________________________________________________________
___________________________________________________________________________________
REVIEW
Simplify.
10) 4x + 2(3x + 4) 11) – 5(x – 8) – 12 12) (4x – 4)2
1 + 4x – 3
Factor.
13) 12x + 16 14) 15x + 30 15) 18x – 45
10
What is pi? Day 4 Classwork
Most people think that the value of pi is 3.14, but that is the rounded decimal for . is an irrational
number. An irrational number never terminates and never repeats.
is a ratio that came from nature. It's the ratio between the circumference of a circle and its diameter,
and it was always there, just waiting to be discovered.
But who discovered it? The Ancient Greek mathematician Archimedes of Syracuse (287-212 BC) is
largely considered to be the first to calculate an accurate estimation of the value of pi.
Below is an discovery activity to find the value of .
Let’s see how close of a decimal you can get to .
Materials needed - Ruler and String.
π = 3.14159265358979323846264338327950288419716…
What do you know about π ? 1) ____________________
2) ____________________
How much String is needed to surround the Circle?
________centimeters
C = π d
How much String is needed to surround the Circle?
________centimeters
C = π d
11
What is pi? Day 4 Classwork
Materials – String and a ruler
Find the Circumference and Diameter in Centimeters. Then solve for π
1) 2)
How much String is needed for How much String is needed for
the Circumference of the Circle? the Circumference of the Circle?
Diameter = ________ cm Diameter = ________ cm
Answer for = ______________ Answer for = ______________
3) 4)
How much String is needed for the How much String is needed for
Circumference of the Circle? the Circumference of the Circle?
Diameter = ________ cm Diameter = ________ cm
12
What is pi? Day 4 Classwork
1) 2)
How much String is needed for the How much String is needed for
Circumference of the Circle? the Circumference of the Circle?
Diameter = ________ cm Diameter = ________ cm
Answer for = ______________ Answer for = _____________
What are some of your Conclusions from the two experiments?
___________________________________________________________________________________
___________________________________________________________________________________
What decimal place in π were you able to get to with your experiment?
___________________________________________________________________________________
Review
Find the missing side of the rectangle given the area:
3) Area = 81x + 18 5) Area = 25x - 40
5
13
2rA
2rA
Circles – Finding Circumference and Area Day 5 Classwork
How do you find the Area of a Circle? How do you find the circumference of a Circle?
Diameter – a line segment that passes through the center of the circle.
Radius – is a line segment that extends from the center of a circle to
any point on the circle.
Area: Formula to find the area of a circle:
To find the Area of a circle follow the following steps:
Write d=_____ r=_______
Write formula:
Substitute for r.
See how the answer should be shown. (Rounding or terms of pi.)
Try this:
a) r = 5 d = ______ b) r = 9 d = _______ c) r = ______ d = 8
d) r = _____ d = 12 e) r = 50 d = _______ f) r = 1 d = ______
g) r = 3 r2
= _____ h) r = 5 r2
= _______ i) r = 1 r2
= _____
Example:
Write r = ______
d = ______
Formula:
Substitute:
Solve:
Answer in terms of π ________
Round answer to the nearest tenth________
diameter
radius
4m
14
dC
dC
Circles – Finding Circumference and Area Day 5 Classwork
Find the area in terms of pi. Find the area and round to the nearest tenth.
1) r = ____ d = _____ 2) r = ____ d = _____
Write formula
3) r = ____ d = _____ 4 ) r = ____ d = _____
Circumference: Formula to find the Circumference of a circle:
To find the Circumference of a circle follow the following steps:
Write d=_____ r=_______
Write formula:
Substitute for d.
See how the answer should be shown. (Rounding or terms of pi.)
Try this:
a) r = 8 d = ______ b) r = 40 d = _______ c) r = ______ d = 24
d) r = _____ d = 12 e) r = 50 d = _______ f) r = 1 d = ______
Example: Find the area Write r = ______
d = ______
Formula:
Substitute:
Solve:
Answer in terms of π ________
Round answer to the nearest tenth________
Ex: Find the Circumference and Area of the following semicircles. (Round to nearest Tenth.)
a. C = ______ b. C = ________
A = ______ A = ________
Diameter is 20 m Radius is 8.2 ft
3m
9m
m
24m
10m
12m
15
Circles – Finding Circumference and Area Day 5 Classwork Find the circumference in terms of pi. Find the circumference and round to the
nearest tenth. 1) r = ____ d = _____ 2) r = ____ d = _____
Write formula
3) r = ____ d = _____ 4) r = ____ d = _____
Try These: Find the Area and the Circumference of the following circles to the nearest tenths place.
5) r = ____ d = _____ 6) r = ____ d = ______
Write formulas:
7) 8) r = ____ d = ____
r = ____ d = _____
M3L16 M3L17
9) Find the area and perimeter of the following: 10) Find the area of the circle below:
3m
7m
4m
m
13m
24m
10m
8m
50m
16
Circles – Finding Circumference and Area Day 5 Classwork
Find the Area and Circumference of the following circles to the nearest tenths place.
1) r = ____ d = _____ 2) r = ___ d =_____
A = C = A = C =
3) r = _____ d = ______ 4) r = ____ d =______
A = C = A = C =
5) How much fencing is needed for a garden that has a diameter of 10 ft? (Round to nearest tenth)
6) How much icing is needed to cover a Cake that has a diameter of 16 inches? (Leave in terms of π)
7) Joan determined the area of the circle below to be 400 𝝅 cm2 but Melinda says the area is 100 𝝅 cm2.
Who is incorrect and why?
Review:
8) Find the missing angle 9) Find the area of these two similar rectangles
9m 6m
12m x
22m
8m
8888
m
18m
6m
1000
1350
700
x
17
2rA
dC
Area vs. Circumference Day 6 Classwork
Sometimes you will be given the Area and be asked to find the Circumference.
**Remember:
Steps:
1) Using the Area formula, plug in the given Area for A
2) Find the Radius by solving for r
3) Find the Diameter and use the Circumference formula to find the Circumference.
Find the Circumference of each in terms of .
1) A = 25 2) A = 4 3) A = 121
Find the Circumference. Rounded to the nearest hundredths place.
4) A = 153.938 m2
5) A = 254.469 cm2
6) A = 28.2743 ft2
Sometimes you will be given the Circumference and be asked to find the Area.
**Remember:
Steps:
1) Using the Circumference formula, plug in the given Circumference for C
2) Find the Diameter by solving for d
3) Find the Radius and use the Area formula to find the Area.
Find the Area of each in terms of .
7) C = 10 8) C = 44 9) C = 8
Find the Area Rounded to the nearest Hundredth.
10) C = 37.6991 m 11) C = 62.8318 cm 12) C = 43.9822 ft
13)The circle below has a diameter of 12 cm. Calculate the area of the shaded region.
18
Area vs. Circumference Day 6 Classwork
2) Harry’s Pizzeria is having a sale on medium and large pizzas. Medium pizzas are 10 inches in
diameter and cost $7.99. Large pizzas have an area of 254 in2
and cost $14.99. Which size pizza
is the better deal? Explain.
__________________________________________________________________________________________
__________________________________________________________________________________________
_____________________________________________________________________
3) If the length of the radius of a circle is doubled, how does that affect the circumference and area?
Explain.
__________________________________________________________________________________________
__________________________________________________________________________________________
_____________________________________________________________________
4) Every year in September, Sue covers her circular pool. Her pool has a diameter of 25 feet. Find
how much covering she will need. Also state if you are finding the Circumference or the Area of
the pool.
5) At a local park, Sara can choose between two circular paths to walk. One path has a diameter of 120
yards, and the other has a radius of 45 yards. How much further can Sara walk on the longer path
than the shorter path if she walks the path once?
Multiple Choice
6) Lana is putting lace trim around the border of a circular tablecloth. The tablecloth has a diameter of
1.2 meters. To the nearest meter, what is the least amount of lace she needs?
A. 3m B. 4m C. 7m D. 8m
7) A graphic artist is designing a company logo with two concentric circles (two circles that share the same
center but have different length radii). The artist needs to know the area of the shaded band between the two
concentric circles. Explain to the artist how he would go about finding the area of the shaded region.
19
Area vs. Circumference Day 6 Classwork
1) Given: A = 36 2) Given: C = 30
Find the Circumference in terms of Find the Area in terms of
3) Given: A = 452.3893m2
4) Given: C = 50.2654cm2
Find the Circumference round to nearest tenth. Find the Area rounded to nearest Tenth.
5) A circular swimming pool has a radius of 15 feet. The family that owns the pool wants to put up a
circular fence that is 5 feet away from the pool at all points.
a) Find the radius of the fenced in area.
b) Find the amount of fencing needed.
6) What is the radius of a circle when the circumference is 16 cm?
7) A circular rose garden needs new sod. The diameter of the garden is 18 feet. How much sod is
needed to cover the rose garden?
8) The front wheel of a high-wheel bicycle from the late 1800s was larger than the rear wheel to
increase the bicycle’s overall speed. The front wheel measured in height up to 60 in. Find the
circumference and area of the front wheel of the high-wheel bicycle.
9) Use the key on the calculator to find the area of a circle whose radius is 5.6m. Which is the better
estimate, 98m2
or 99m2
? Explain.
___________________________________________________________________________________
___________________________________________________________________________________
10) The circumference of a circle is 24𝜋 cm. What is the exact area of the circle?
Review
Determine if the following is an equation, expression, or inequality.
10) x = 7 11) 3x + 4y2 12) x > -2 13) 5x 10
20
Using a Protractor Day 7 Classwork
A Protractor is used to measure the degrees of an angle or draw an angle.
Always be careful which numbers to use on the Protractor depending on which way the angle is
opening up.
Types of Angles: Name of Angle Definition Picture
Right Angle
Obtuse Angle
Acute Angle
Straight Angle
Reflex Angle
21
Using a Protractor Day 7 Classwork State the type of angle and measure the degrees of each angle using the protractor.
1) _____________ angle
_____________ degrees
2) ____________angle
_____________ degrees
3) ___________ angle
_____________ degrees
4) _____________ angle
_____________ degrees
5) ___________ angle
_____________ degrees
6) ____________ angle
_____________ degrees
Draw angles using the protractor
1) 95 degrees
2) 160 degrees
3) 55 degrees
4) 250 degrees
5) 45 degrees 6) 90 degrees
22
Using a Protractor Day 7 Classwork
State the type of angle and measure the degrees of each angle using the protractor.
1)
_____________ degrees
2)
_____________ degrees
3)
_____________ degrees
4)
_____________ degrees
5)
_____________ degrees
6)
_____________ degrees
Draw angles using the protractor
1) 105 degrees 2) 140 degrees
3) 85 degrees 4) 200 degrees
5) 25 degrees 6) 180 degrees
23
Classifying and Drawing Quadrilaterals Day 8 Classwork
Quadrilateral: ______________________________
A square is a quadrilateral with four equal parallel sides and four right angles.
A rectangle is a quadrilateral with two sets of parallel sides and four right
angles.
A trapezoid is a quadrilateral with exactly one pair
of parallel sides.
A rhombus is a quadrilateral with four equal sides and two sets of parallel sides.
A parallelogram is a quadrilateral with two
pairs of parallel sides. The opposite sides
and angles are congruent.
Measure and label each angle with your protractor.
Add the 4 angles in each figure to come up with the sum of the interior angles.
Sum of Angles = _____________ Sum of Angles = ________
**All Quadrilateral’s interior angles have the same sum of ________ degrees!
24
Classifying and Drawing Quadrilaterals Day 8 Classwork Using the protractor, draw a Quadrilateral with the following 4 angle measures.
1) 90 0 , 90 0 , 90 0 , 90 0 2) 150 0 , 30 0 , 150 0 , 30 0
3) 1300, 50
0, 120
0, 60
0 4) 100
0, 50
0, 130
0, 80
0
5) A quadrilateral has angles measuring 500, 70
0, and 100
0. What is the measure of the fourth angle?
A. 400 B. 120
0 C. 80
0 D. 140
0
6) Which expression can be used to find the measure of angle m in this quadrilateral?
1330
700
A. 360 + (133 – 84 – 70)
B. 360 + (133 + 84 + 70)
840
m C. 360 – (133 – 84 – 70)
D. 360 – (133 + 84 + 70)
25
Classifying and Drawing Quadrilaterals Day 8 Classwork Choose the best answer for each question.
1) What is the best name of this quadrilateral?
A. Square
B. Rectangle
C. Trapezoid
D. Parallelogram
E. Rhombus
2) What is the best name of this quadrilateral?
A. Square
B. Rectangle
C. Trapezoid
D. Parallelogram
E. Rhombus
3) What is the measure of x ?
Draw a Quadrilateral using the protractor with the following measurements.
4) 700, 110
0, 70
0, 110
0
5) 600, 120
0, 40
0, 140
0
3 in.
3 in.
100 0 125 0
80 0 x
26
Classifying and Drawing Triangles Day 9 Classwork
Types of Triangles: Name of Triangle Definition Picture
Acute Triangle
Right Triangle
Obtuse Triangle
Scalene Triangle
Isosceles Triangle
Equilateral
Triangle
**The sum of the measures of the angles of any triangle is 180 0 .
Find the measure of the missing angle. (Not drawn to scale)
1. 300
2. x 3. x
700
x 410
700
700
27
Classifying and Drawing Triangles Day 9 Classwork
Drawing Triangles: Draw a triangle with the following three angle measures: 40 0 , 80 0 , 60 0
Steps: ◦ Draw a ray and create one of the angles (40 0 ) ◦ Over extend the line of the 40 0 ◦ Place a vertex on that line to draw your next angle (80 0 ) ◦ The new ray must intersect the original ray to create a triangle ◦ Check to make sure the last angle created is 60 0 Draw a triangle with the following angle measures:
1) 60 0 , 50 0 , 70 0 2) 100 0 , 40 0 , 40 0
3) 75 0 , 90 0 , 15 0 4) 80 0 , 30 0 , 70 0
5) Draw an isosceles triangle △𝐴𝐵𝐶. Begin by drawing ∠𝐴 with a measurement of 80°. Use the rays of ∠𝐴 as the equal legs of the triangle. Choose a length of your choice for the legs and use your compass to mark off each leg. Label each marked point with 𝐵 and 𝐶. Label all angle measurements.
28
Classifying and Drawing Triangles Day 9 Classwork
Find the missing angle measure of the following triangles:
1) 2) 3) What is the third angle
x x of a triangle that has two
measures of 110 and 35 degrees?
50 0 70 0 39 0 88 0
Construct a Triangle with the following angle measurements.
4) 0
0
0
20
70
90
BAC
BCA
ABC
5) 0
0
0
45
45
90
NMO
NOM
MNO
6) Explain the steps you would take to create a triangle if you were given the
measures of all three angles.
__________________________________________________________________
__________________________________________________________________
7) Determine all possible measurements in the following triangle and use your tools to create a copy of it.
29
Drawing Triangles by SAS and ASA Day 10 Classwork
You have already constructed Triangles by 3 angles (AAA) earlier in this unit.
We will now construct a Triangle by SAS (Side Angle Side)
and ASA (Angle Side Angle)
The middle fact is where you must start your construction:
When given ASA – Start by constructing the Side given.
When given SAS – Start by constructing the Angle given.
Example 1:
Draw a triangle that satisfies the set of conditions below: Steps:
(ASA) 800 , 3cm, 40
0 - Draw a line segment 3cm long.
- Each endpoint of the line
segment is the vertex of the
two angles given.
- One vertex draw a 800
angle
from.
- The other vertex draw a 400
angle from.
- Connect the new rays to form
a triangle.
Example 2:
Draw a triangle that satisfies the set of conditions below: Steps:
(SAS) 4cm, 1250 , 5cm - Draw a line segment
- At one end point draw a
1250
angle
- Measure each of the line
segments (4cm and 5cm)
- Connect end of the 4cm to the
end of the 5cm line segment.
**Always check the remaining angles so that every triangle has a sum of 180 0
30
Drawing Triangles by SAS and ASA Day 10 Classwork
Examples:
Draw a triangle that satisfies the set of conditions below:
1) (SAS) 5cm, 85 0 , 2cm
2) (SAS) 3cm, 400 , 4cm
3) (ASA) 500, 2cm, 50
0
4) (ASA) 800, 4cm, 70
0
5) Draw a triangle with one obtuse angle and no congruent sides.
31
Drawing Triangles by SAS and ASA Day 10 Classwork
Draw a triangle that satisfies the set of conditions below:
1) (SAS) 4cm, 500, 6cm 2) (ASA) 70
0, 3cm, 50
0
3) (ASA) 800, 5cm, 30
0 4) (AAA) 70
0, 80
0, 30
0
5) Draw a triangle with one right angle and no congruent sides.
6) Draw a triangle with one obtuse angle and two congruent sides.