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2-Dimensional Figures

Name: _________________

Teacher: _______________

Grade: ________

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.

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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.

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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.