1. Geometry & Measurement
Math 6

2. Polygons
Geometry & Measurement

3. Polygons
Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed".

4. Polygons
Types of Polygons A regular polygon has all angles equal and all sides equal otherwise it is irregular

5. Polygons Types of Polygons
A convex polygon has no angles pointing inwards.
If any internal angle is greater than 180° then the polygon is concave.

6. Polygons Triangle (3 Sides) Names of Polygons Quadrilateral (4 Sides)
Pentagon (5 Sides)
Hexagon (6 Sides)
Octagon (8 Sides)
Decagon (10 Sides)

7. Polygons
ment/polygons/

8. Angles
Geometry & Measurement

9. Angles
There are 4 parts to an angle that you need to know

10. Angles
Arm
Angle Measure (inside)
Vertex
Arm

11. Angles Angles measure the amount of turn
Angles are measured in degrees
The symbol for degree is 0 (for example 900)

12. Angles There are 360 degrees in one complete circle Benchmark Angles:
Angles such as 45°, 90°, 180° that can be used as references.

13. Angles Names (Classification) of Angles
An Acute Angle is less than 90° (0° -90°)

14. Angles Names (Classification) of Angles A Right Angle is exactly 90°
Note the special symbol. If you see this, you are being told it is a right angle.

15. Angles Names (Classification) of Angles
An Obtuse Angle is more than 90° but less than 180°

16. Angles Names (Classification) of Angles
A straight angle is exactly 180 degrees

17. Angles Names (Classification) of Angles
A Reflex Angle is more than 180° but less than 360°

18. Angles
Remember to look carefully at which angle you are being asked to name
The reflex angle is the larger angle. It is more than 180° but less than 360°

19. Angles
What type of angle is this?

20. Angles
What type of angle is this?

21. Angles
What type of angle is this?

22. Angles
What type of angle is this?

23. Angles
What type of angle is this?

24. Angles
What type of angle is this?

25. Angles Measuring Angles
a protractor, is used to measure angles in degrees

26. Angles

31. Angles Naming an angle Use three letters to name an angle.
Always put the vertex in the middle.
The name for this angle is GAW or WAG or A A G W

35. Angles http://www.softschools.com/math/geometry/angles/a ngle_types/
ement/angles/

36. Angles
Constructing Angles
a protractor, is used to construct angles

37. Angles Constructing Angles You will create an angle measuring 370
1. Draw a line segment (Using a ruler). Make a dot at one end.

38. Draw a line segment (Using a ruler). Make a dot at one end

39. Angles Constructing Angles
2. Center the protractor over the dot. Line it up so that the segment passes through the 0 degree mark.

40. Center the protractor over the dot
Center the protractor over the dot. Line it up so that the segment passes through the 0 degree mark.

41. Angles Constructing Angles
3. Find the number that matches the size of the angle you wish to draw, and draw a dot at that number.

42. Find the number that matches the size of the angle you wish to draw, and draw a dot at that number.

43. Angles Constructing Angles
4. Connect the dots with your straight edge.

44. Connect the dots with your straight edge

45. Angles
Constructing Angles
5. Label the angle.

46. Label the angle
370

47. Angles of Polygons
Geometry & Measurement

48. Angles of Polygons
Without using a protractor, what is the measure of this angle?
How do you know?

49. Angles of Polygons
Without using a protractor, what is the measure of angle y?
How do you know?

50. Angles of Polygons
A triangle has three sides and three angles

51. Angles of Polygons
A triangle has three sides and three angles

52. Angles of Polygons
A triangle has three sides and three angles

53. Angles of Polygons A triangle has three sides and three angles
The three interior angles always add to 180°

54. Angles of Polygons

55. Angles of Polygons Formula for the sum of the angles in a triangle is:
∠A + ∠ B + ∠C = 180°
= 1800

56. Angles of Polygons
How can we solve this? ∠A + ∠ B + ∠ C = 180° ∠A = 180° ∠A = 180° So ( = 30) ∠A = 30°
We can use the sum of the angles in a triangle to find the measure of the third angle in this triangle.

57. Angles of Polygons
You Try

58. Angles of Polygons
You Try

59. Angles of Polygons
You Try

60. Angles of Polygons
You Try

61. What do these polygons have in common?

62. What can you deduce about the interior angles of these quadrilaterals?

63. Angles of Polygons
The sum of the interior angles in a quadrilateral is the same for any quadrilateral.
A diagonal divides any quadrilateral into 2 triangles.

64. Angles of Polygons
Formula for the sum of the angles in a quadrilateral is:
∠A + ∠ B + ∠C + ∠ D = 360°
45°+ 135°+ 120°+ 60°= 360°

65. Angles of Polygons W
How can we solve this? ∠X + ∠ W + ∠ Y + ∠ Z = 360° ∠X + 75°+ 110°+ 100°= 360° ∠X = 360° So ( = 75) ∠X = 75° Z Y
We can use the sum of the angles in a triangle to find the measure of the third angle in this triangle.

66. Angles of Polygons
You Try

67. Angles of Polygons
You Try

68. Triangles
Geometry & Measurement

69. Triangles
There are 2 ways to classify triangles. Based on these images can you figure what the 2 ways are?

70. Triangles
There are three classifications given to triangles based on how many sides are equal.
There are three classifications given to triangles based on the size of the interior angle(s).

71. Triangle
What Type of Angle? Acute Triangle: A triangle that has all angles less than 90°

72. Triangle
What Type of Angle? Obtuse Triangle: A triangle that has an angle greater than 90°

73. Triangle
What Type of Angle? Right Triangle: A triangle that has a right angle (90°)

74. Triangle
What Type of Sides? Equilateral Triangle: A triangle with all three sides of equal length. All the angles are 60°

75. Triangle
What Type of Sides? Isosceles Triangle: A triangle with two equal sides. The angles opposite the equal sides are also equal.

76. Triangle
What Type of Sides? Scalene Triangle: A triangle with all sides of different lengths. No sides are equal and no angles are equal.

77. Triangles
ement/typesoftriangles/

78. Perimeter of Polygons
Geometry & Measurement

79. Perimeter of Polygons Formula:
A special type of equation that shows the relationship between different variables.
It’s like a recipe.
Variable:
A symbol for a number we don't know yet. It is usually a letter like x or y.

80. Perimeter of Polygons The distance around a two-dimensional shape.
It is a measure of length

81. Perimeter of Polygons The perimeter of this polygon is
P = 7.6m

82. Perimeter of Polygons
You try
P =
P =
P = 18ft

83. Perimeter of Polygons
You try
P =
P =
P = 26cm

84. Perimeter of Polygons You try P = P = 10 + 3 + n + s + 4 + 5
P = 30cm

85. Perimeter of Polygons These are the formulas irregular polygons:
# of sides
formula
triangle
a + b + c
quadrilateral
a + b + c + d
pentagon
a + b + c + d + e
hexagon
a + b + c + d + e + f
oxtogon
a + b + c + d + e + f + g + h

86. Perimeter of Polygons Regular polygons have specific formulas
Try to create a formula for a square

87. Perimeter of Polygons
Regular polygons have specific formulas The formula for a square is: P = 4s P = 4(7) which means 4 x 7 P = 28

88. Perimeter of Polygons
Regular polygons have specific formulas Try to create a formula for a Hexagon 5

89. Perimeter of Polygons
Regular polygons have specific formulas The formula for a hexagon is: P = 6s P = 6(5) P = 30 5

90. Perimeter of Polygons Regular polygons have specific formulas
Try to create a formula for a rectangle

91. Perimeter of Polygons Regular polygons have specific formulas
The formula for a rectangle is:
P = 2(l + w)
P = 2 (10 + 6)
P = 2 (16)
P = 32

92. Perimeter of Polygons Regular polygons have specific formulas
The formula for a rectangle is:
P = 2l + 2w
P = 2 x x 6
P =
P = 32

93. Area of Polygons
Geometry & Measurement

94. Area of a Polygon Area is the size of a surface! (surface area)
It helps to imagine how much (paint, wallpaper, wrapping paper) it would take to cover the shape if you were using squares.

95. Area of Polygons
To find out how much shape is inside we can count the squares.
The area of the shape is 182cm2.

96. Area of Polygons What is cm2?
The area equal to a square that is 1 centimeter on each side.
The symbol is cm2
There is also m2
There is in2
There is km2

97. What is the area of each of these shapes?
18cm2
20cm2
30cm2
14cm2
22cm2

98. Area of a Polygon
Counting each square can take a long time. You can remove the squares and use the dimensions
The rectangle has a length of 8cm
The rectangle has a width of 6cm
To work out the number of squares we multiply 8 by 6.
Length (8cm)
Width (6cm)

99. Area of Polygons Regular polygons have specific formulas for area
Try to create a formula for a rectangle

100. Area of Polygons
A = b * h
A = 11 * 4
A = 44cm2 11 4

101. Area of Polygons
You Try:
A = b * h
A = 7 * 5
A = 35cm2
7cm
5cm

102. Area of Polygons
You Try:
A = b * h
A = 12.5 * 2
A = 25cm2
12.5cm
2cm

103. Area of Polygons 4cm You Try: first find the missing #’s
split it up into two rectangles.
2cm
4cm
10cm
8cm
8cm

104. Area of Polygons You Try: A1 = b * h A = 10 * 4 A = 40cm2 A2 = b * h

105. Area of Polygons Regular polygons have specific formulas for area
Try to create a formula for a triangle
Hint – what shape does 2 triangles make?

106. Area of Polygons Regular polygons have specific formulas for area
The formula for a rectangle is A= b*h
A triangle is of a rectangle
A = 𝑏ℎ 2 or bh
Multiply base and height
Divide by 2

107. Area of Polygons
You Try:
A = 𝑏ℎ 2
A = 6∗4 2
A = 24 2
A = 12cm2

108. Area of Polygons
You Try:
A = 𝑏ℎ 2
A = 14∗5 2
A = 70 2
A = 35cm2

109. Area of Polygons Parallelogram: is a quadrilateral (four sides)
opposite sides are parallel
opposite sides are equal in length
opposite angles are equal (A & C)

110. Area of Polygons Parallelogram: The area is base x height:
Area = b * h

111. Area of Polygons
Parallelogram:
Here’s why

112. Area of Polygons Parallelogram: The area is base x height:
Area = b * h

113. Area of Polygons Parallelogram: The area is base x height:
Area = b * h

114. Area of Polygons Trapezoids is a quadrilateral (four sides)
has a pair of opposite sides parallel (not equal length)
Has two sides not parallel or equal length

115. Area of Polygons Trapezoids
Regular polygons have specific formulas for area
Try to create a formula for a trapezoid

116. Area of Polygons Trapezoids
Regular polygons have specific formulas for area
A = 𝑎+𝑏 ∗ℎ 2

117. Area of Polygons Trapezoids
Regular polygons have specific formulas for area
A = 𝑎+𝑏 ∗ℎ 2

118. Area of Polygons Trapezoids A = 𝑎+𝑏 ∗ℎ 2 A = 5+10 ∗8 2 A = 15 ∗8 2
A = 60 cm2 10
5cm 8 8 10 5

119. Area of Polygons You Try: A = 𝑎+𝑏 ∗ℎ 2 A = 5+8 ∗4 2 A = 13∗4 2
A = 26 m2

120. Area of Polygons You Try: A = 𝑎+𝑏 ∗ℎ 2 A = 9+7 ∗5 2 A = 16∗5 2
A = 40 in2

121. Area of Polygons
ment/areaofpolygons/

122. Volume of Polygons
Geometry & Measurement

123. Volume of Polygons
Volume is the amount of “space” an object occupies.

124. Area of Polygons Cubic Unit: a measure of volume.
It is equal to the volume of a cube, which is 1 unit high, 1 unit wide and 1 unit long.
The symbol is unit3
Examples: cm3, in3

125. Volume of Polygons Simplified:
Volume is the amount of little cubes that will fill up the big cube.

126. Volume of Polygons A cube has 3 different dimensions: Length Width
Height

127. Volume of Polygons
The formula of a cube is: V = l * w * h

128. Volume of Polygons How many cubes does this rectangle have?
L = 10, W = 1, H = 1
V = l * w * h
V = 10 * 1 * 1
V = 10 units3

129. Volume of Polygons How many cubes does this rectangle have? L = 10
V = l * w * h
V = 10 * 5 * 1
V= 50 units3

130. Volume of Polygons How many cubes does this rectangle have? L = 10
V = l * w * h
V = 10 * 5 * 6
V= 300 units3

131. Volume of Polygons
You Try
V = l * w * h
V = 3 * 3 * 3
V = 27 units3

132. Volume of Polygons
You Try
V = l * w * h
V = 3 * 3 * 4
V = 36 cm3

133. Volume of Polygons
You Try
V = l * w * h
V = 10 * 3 * 4
V = 120 in3

134. Volume of Polygons A Side Note:
Volume, whether liquid or solid, is a measure of space.
Solid volume is measured using cubic units.
Liquid volume is most often measured using liters.

135. Volume of Polygons
A Side Note:
1 mL is equivalent to 1 cm3

136. Volume of Polygons Keep Calm and Pass Your Math Test! Volume HW Review
Practice Test
Test