Practice QuizPolygons, Area

Perimeter, Volume

1 Two angles of a hexagon measure 140° each. The other four angles are equal in measure. What is the measure of each of the other four equal angles, in degrees?

x

140 140

x

x x

Step 1: Find the sum of interior angles in a hexagon. Number of sides = 6

Number of sides – 2 = 6 – 2 = 4

Multiply 4 by 180 = 4(180) = 720

1 Two angles of a hexagon measure 140° each. The other four angles are equal in measure. What is the measure of each of the other four equal angles, in degrees?

x

140 140

x

x x

Step 1: Find the sum of interior angles in a hexagon. 720Step 2: Set up equation by letting sum of angles equal 720.x + x + x + x + 140 + 140 = 720

4x + 280 = 720– 280 – 280

4x = 440x = 110

Measure of each the four equal angles is 110

In trapezoid ABCD, AB = CD. What is the value of x?

Isosceles Trapezoid

A and Bsupplementary75° + x = 180°

x = 105°

Method #1

2

–75 –75

Sum of all angles = 360°

x + x + 75 + 75 = 360

x = 105

Method #2

2x + 150 = 360

2x = 210

2 In trapezoid ABCD, AB = CD. What is the value of x?

–150 –150

3 In quadrilateral DEFG, is parallel to . What is the measure of F?

DGEF

Trapezoid

F and Gsupplementary

x + 10 + x = 180

x = 85

–10 –102x + 10 = 180

(x+10)

xG

FE

D2x = 170 F = x + 10

= 85 + 10 = 95

4 In the figure, what is the value of x ?

Step 1: Solve for c using pythagorean theorem.

a2 + b2 = c2

3

3

x

7c 22 23 7 c

9 + 7 = c2

16 = c2

4 = c

216 c

4 In the figure, what is the value of x ?

Step 2: Solve for x using pythagorean theorem.

3

3

x

74

a2 + b2 = c2

9 + 16 = x2

25 = x2

5 = x

32 + 42 = x2

225 x

5 In the figure, what is the length of ?

Step 1: Solve for ? using pythagorean theorem.

a2 + b2 = c2

3

B

?9 + 16 = ?2

25 = ?2

5 = ?

AB

A

C

D4

13

x32 + 42 = ?2

225 ?x is length of AB

5

Step 2: Solve for x using pythagorean theorem.

a2 + b2 = c2

x2 + 25 = 169

x = 12

x2 + 52 = 132

In the figure, what is the length of ?AB

3

B

5

C

D4

13

x

A

–25 –25x2 = 144

x is length of AB2 144x

6 Find the value of each interior anglefor a regular polygon with 20 sides.

Step 1: Find sum of the interior angles in a regular polygon with 20 sides. Number of sides = 20 Number of sides – 2 = 20 – 2 = 18

Multiply 18 by 180 = 18(180) = 3240

Step 2: Find value of each interior angle. Divide sum by number of sides, 20.

3240 20

= 162

7 A regular octagon is shown. What is the measure, in degrees, of X?

Number of sides = 8 Number of sides – 2 = 8 – 2 = 6

Multiply 6 by 180 = 6(180) = 1080

Step 2: Find value of each interior angle, X. Divide sum by number of sides, 8.

1080 8 = 135

Step 1: Find sum of the interior angles in the regular octagon.

X

8 In the figure, . What is the value of x.

AF GD

HCG = DCEVertical Angles

HCG = 15DCE = 15 15

D = 50Corresponding

Angles

D

50

K

K and DSupplementary

Angles

K + D = 180K + 50 = 180

K = 130

8 In the figure, . What is the value of x.

AF GD

15 D

50

130The sum of the angles in ∆CDEis equal to 180 x + 15 + 130 = 180

x + 145 = 180x = 35

In the figure, VW = WX = VX = XY = YZ = XZ. If VZ = 12, what is the perimeter of the triangle VWX?

VZ = 12VX = 6VW = 6WX = 6

Perimeter VWX = 6 + 6 + 6 = 18

9

10

The perimeter of an isosceles triangle is 20 inches, its base measures 8 inches. Find the length of each of its equal sides in inches.

8

xx

Perimeter = 20 inches

x + x + 8 = 202x + 8 = 20

–8 –82x = 12

x = 6

x = length of each equal side

11

If each of the equal sides of an isosceles triangle is 10, and the base is 16, what is the area of the triangle?

16

1010

12

A bh

Base (b) = 16

Find height (h)

h

Use Pythagorean Theorem

a2 + b2 = c2

8

h2 + 82 = 102

h2 + 64 = 100

h2 = 36h = 6

–64 –64

11

If each of the equal sides of an isosceles triangle is 10, and the base is 16, what is the area of the triangle?

16

1010

12

A bh

Base (b) = 16

Find height (h)

6

8

h = 612

A bh1 16 62

1 962 1

1 962

A

962

A = 48

12

In the figure, E is the midpoint of side CB of rectangle ABCD, and x = 45°. If AB is 3 centimeters, what is the area of rectangle ABCD, in square centimeters?

x

3

=45

3

∆DCE is isoscelesCD CE

3CD 3CE

45

453 3

6Area of rectangle ABCDLength Width = 6

3= 18

If the area of a right triangle is 16, the length of the legs could be

13

h

b

A. 8 and 2

B. 12 and 4

C. 10 and 6

D. 20 and 12

E. 32 and 1

1 8 22

A 1 162

= 8

= 241 12 42

A 1 482

= 301 10 62

A 1 602

= 1201 20 122

A 1 2402

= 161 32 12

A 1 322

1Find Area: 2

A bh

14

In the figure, right triangle ABC is contained within right triangle AED. What is the ratio of the area of AED to the area of ABC?

∆ABC is Isosceles

45

45

45

8

AC BC

∆AED Big TriangleA = 45 D = 90 E = 45

∆AED is Isosceles

ED AD

8ED 8AD

14

In the figure, right triangle ABC is contained within right triangle AED. What is the ratio of the area of AED to the area of ABC?

Area of ∆ABC Small Triangle

45

45

45

8

Area of ∆AED Big Triangle

12

A bh 612

6 1 362

= 18

12

A bh 812

8 1 642

= 32

Area RatioArea

AEDABC

3218

169

The figure above shows a square region divided into four rectangular regions, three of which haveareas 5x, 5x, and x2, respectively. If the area of MNOP is 64, what is the area of square QROS?

15

x

x

x

x

5

5 5

5Area of square QROS

Length Width

= 5 5= 25

In the figure, CDE is an equilateral triangle and ABCE is a square with an area of 1.What is the perimeter of polygon ABCDE?

16

ABCE is a squarewith an area of 1

Area = s2

1 = s2

1 = s

1

1

1

1

1

1

Perimeter of ABCDE1 + 1 + 1 + 1 + 1 = 5

One-third of the area of a square is 12 square inches. What is the perimeter of the square, in inches?

17

1 123

A

31 13

3 2A

A = 36

A = s2s2 = 36s = 6

6 6

6

6Perimeter = 4(6)

Perimeter = 24

All the dimensions of a certain rectangular solid are integers greater than 1. If the volume is 126 cubic inches and the height is 6 inches, what is the perimeter of the base?

V = lwh126 = l w 6

21 = l w

V = Volume

(Base)

l

w

h

Base

6

26 66 6

1 l w

l = 3 or 7w = 3 or 7

Perimeter of Base

2l + 2w= 2(7) + 2(3)= 14 + 6= 20

18

A rectangular solid has a square base. The volume is 360 cubic inches and the height is 10 inches. What is the perimeter of the base?

19

V = lwh360 = l w 10

36 = l w

V = Volume

(Base)

l

w

h

Base

10

l = 6 and w = 6

Perimeter of Base

4(s)= 4(6)

= 24360 l·w·10=10 10

20

Cube A has an edge of 2. Each edge of cube A is increased by 50%, creating a second cube B. What is the ratio of the volume of cube A to cube B?

2

22

Cube A

50% of 2 = .50 2

= 1

50% increase = 2 + 1 = 3

Cube B

3

33

V = side3

V = 23 = 8 V = side3

V = 33 = 27

Volume Cube ARatioVolume Cube B

827

21

Cube A has an edge of 2. Each edge of cube A is increased by 50%, creating a second cube B. The surface area of cube B is how much greater than the surface area of cube A?

2

22

Cube A Cube B

3

33

SA = 6s2

= 622

SA = Surface Area

= 64= 24

SA = 6s2

= 632= 69= 54

SA Cube B SA Cube A–

54 – 24 = 30

22 How many wooden toy cubes with a 3-inch edge can fit in a rectangular container with dimensions 3 inches by 21 inches by 15 inches?

V = side3

V = 33 = 273

V = lwhV = 3 21 15V = 945

3

15

21

Volume ofLarge Container

Volume ofOne Toy Cube

V = Volume

Find number of toy cubes in rectangular (large) container

945 27 = 35

23 If you assume that there is no wasted ice, how many smaller rectangular block ice cubes, dimensions 234, can be cut from two large blocks of ice? The size of each block of ice is shown below.

V = l w hV = 2 3 4

V = l w hV = 4 10 6V = 2406

4

10

V = Volume Ice Cube

V = 24

Volume ofLarge Block of Ice

Volume ofOne Ice Cube

Find number of ice cubes in one large block of ice

240 24 = 10

Ice cubes in two large

blocks of ice2(10) = 20

24 If cube B has an edge three times that of cube A, the volume of cube B is how many times the volume of cube A?

V = VolumeStrategy: Substitute a number for each cube edge.

131= 3

Cube BCube A

V = s3

V = 13

V = 1

V = s3

V = 33

V = 27

Volume Cube B ÷ Volume Cube A

= 27 ÷ 1 = 27

25 The surface areas of the rectangular prism are given. If the lengths of the edges are integers, what is the volume in cubic inches?

28 = 1 28

24 sq in.

28 sq in.

42 sq in.

Strategy: Use trial and error with different combinations of numbers to find the area of each face.

28

1

28

42 = 28 ?

?

No integer factor for 42

Combination #1 NO

25 The surface areas of the rectangular prism are given. If the lengths of the edges are integers, what is the volume in cubic inches?

24 sq in.

28 sq in.

42 sq in.

Strategy: Use trial and error with different combinations of numbers to find the area of each face.

14

2

14

3

28 = 2 1442 = 14 3

38

24 = 3 8

Combination #2

28 = 1 2842 = 28 ?

No integer factor for 42

Combination #1

Parallel line segments not

equal

NO

NO

25 The surface areas of the rectangular prism are given. If the lengths of the edges are integers, what is the volume in cubic inches?

24 sq in.

28 sq in.

42 sq in.

Strategy: Use trial and error with different combinations of numbers to find the area of each face.

7

4

7

6

28 = 4 742 = 7 6

64

24 = 6 4

Combination #3 YES

Volume = l · w · hVolume = 4 · 6 · 7Volume = 168 cubic inches