Math 8 Unit 8 Polygons and Measurement Strand 4: Concept 4 Measurement Strand 4: Concept 1 Geometric Properties PO 2. Draw three-dimensional figures by applying properties of each PO 3. Recognize the three-dimensional figure represented by a net PO 4. Represent the surface area of rectangular prisms and cylinders as the area of their net. PO 5. Draw regular polygons with appropriate labels PO 1. Solve problems for the area of a trapezoid. PO 2. Solve problems involving the volume of rectangular prisms and cylinders. PO 3. Calculate the surface area of rectangular prisms or cylinders. PO 4. Identify rectangular prisms and cylinders having the same volume.
Key Terms Def: Polygon: a closed plane figure formed by 3 or more segments that do not cross each other. Def: Regular Polygon: a polygon with all sides and angles that are equal. Def: Interior angle: an angle inside a polygon Def: Exterior angle: an angle outside a polygon
Triangle 3 Sides 3 Angles Sum of Interior Angles 180 Each angle measures 60 if regular.
Quadrilateral 4 Sides 4 Angles Sum of Interior Angles 360 Each angle measures 90 if regular.
Pentagon 5 Sides 5 Angles Sum of Interior Angles 540 Each angle measures 108 if regular.
Hexagon 6 Sides 6 Angles Sum of Interior Angles 720 Each angle measures 120 if regular.
Heptagon 7 Sides 7 Angles Sum of Interior Angles 900 Each angle measures if regular.
Octagon 8 Sides 8 Angles Sum of Interior Angles 1080 Each angle measures 135 if regular.
Nonagon 9 Sides 9 Angles Sum of Interior Angles 1260 Each angle measures 140 if regular.
Decagon 10 Sides 10 Angles Sum of Interior Angles 1440 Each angle measures 144 if regular.
Formula:
Example: Find the sum of the interior angles in the given polygon. a. 14-gon b. 20-gon 180(n-2) Total = 2160 180 ● (14-2) 180(n-2) Total = 3240 180 ● (20-2)
Example: Find the measure of each angle in the given regular polygon. a. 16-gon 180(n-2) Total = 2520 180 ● (16-2) 180(n-2) Total = 1800 180 ● (12-2) b. 12-gon 2520 ÷ 1800 ÷
Example: Find the length of each side for the given regular polygon and the perimeter. a.) rectangle, perimeter 24 cm b.) pentagon, 55 m 24 ÷ 4 6 cm 55 ÷ 5 11 m
Formula:
Example: Find the length of each side for the given regular polygon and the perimeter. d. heptagon, 56 mm c.) nonagon, 8.1 ft 8.1 ÷ ft 56 ÷ 7 8 mm
Perimeter Any shape’s “perimeter” is the outside of the shape…like a fence around a yard. Evil mathematicians have created formulas to save you time. But, they always change the letters of the formulas to scare you!
Perimeter Triangles have 3 sides…add up each sides length =24 The Perimeter is 24 To calculate the perimeter of any shape, just add up “each” line segment of the “fence”.
Perimeter A square has 4 sides of a fence =48
Regular Polygons Just add up EACH segment 10 8 sides, each side 10 so =80
Area Area is the ENTIRE INSIDE of a shape It is always measured in “squares” (sq. inch, sq ft)
Different Names/Same idea Length x Width = Area Side x Side = Area Base x Height = Area
Notes 3-D Shapes Base: Top and/or bottom of a figure. Bases can be parallel. Edge: The segments where the faces meet. Face: The sides of a three-dimensional shape. Nets: Are used to show what a 3-D shape would look like if we unfolded it.
Prisms Rectangles Have Rectangles for faces Named after the shape of their Bases
More Nets
by D. Fisher
Vertices (points) Edges (lines) Faces (planes) The base has 3 sides.
Vertices (points) Edges (lines) Faces (planes) The base has sides. 4
Vertices (points) Edges (lines) Faces (planes) The base has sides.5
Vertices (points) Edges (lines) Faces (planes) The base has sides. 6
Vertices (points) Edges (lines) Faces (planes) The base has sides.8
Pyramids Triangles Have Triangles for faces Named after the shape of their bases.
By D. Fisher
Vertices (points) Edges (lines) Faces (planes) The base has 3 sides.
Vertices (points) Edges (lines) Faces (planes) The base has sides.4
Vertices (points) Edges (lines) Faces (planes) The base has sides.5
Vertices (points) Edges (lines) Faces (planes) The base has sides.6
Vertices (points) Edges (lines) Faces (planes) The base has sides.8
Name Picture Base Vertices Edges Faces Triangular Pyramid Square Pyramid Pentagonal Pyramid Hexagonal Pyramid Heptagonal Pyramid Octagonal Pyramid Any Pyramid n n + 1 2n Draw it No picture
Name Picture Base Vertices Edges Faces Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism Heptagonal Prism Octagonal Prism Any Prism n 2n 2n3n n + 2 Draw it No picture
Cylinder Circles Circles for bases Rectangle for side
Points of View View point is looking down on the top of the object. View point is looking up on the bottom of the object. View point is looking from the right (or left) of the object.
Front View Top View Side View Example 1 : Bottom View Bottom Front Side Front Top
Example 2 : Top H D Front View Left View
Example 3 : Top view Left View Front View
Example 4 Left View Top View Bottom View
Surface Area Surface Area: the total area of a three-dimensional figures outer surfaces. Surface Area is measured in square units (ex: cm 2 )
Rectangular Prism SA=2lw +2lh + 2wh ll h hhw w h w w l
1. Find the surface area. SA=2lw +2lh + 2wh W L H SA=2 4 4 8 2 SA= SA= 112 cm 2
2. Find the surface area of a box with a length of 6 in, a width of 6 inches and a height of 10 inches. SA=2lw +2lh + 2wh SA=2 6 6 6 10 SA= SA= 312 cm 2
Cylinder r h r h r SA = r 2 + r 2 + hC C=2 r A=hC So A=2 rh SA =2 r rh
Examples: 1. Find the surface area. SA =2 r rh SA = 2 (5) (5)(20) SA = 2 100 SA = 50 SA = 250 = 785 cm 2
2. Find the surface area of a cylinder with a height of 5 in and a diameter of 18 in. SA =2 r rh SA = 2 (9) (5)(18) SA = 2 90 SA = 162 SA = 342 = in 2
Volume Volume: The amount of space inside a 3D shape. Volume is measure in cubic units (ex: cm 3 )
Rectangular Prism V=LWH V = 8 4 2 V = 64 cm 3
Cylinder V= r 2 h V = 2 (5) 2 (20) SA = 1000 = 3140 cm 3 V = 2 25 20 V = 2 500
Triangular Prism V= ½ LWH V = ½ 24 49 10 V = 5, 880 cm 3
Surface Area or Volume Covering a Triangular speaker box with carpet? Surface Area Filling a triangular speaker box with foam? Volume Filling a triangular box with M n M’s? Volume
Surface Area or Volume Painting the outside of a triangular prism? Surface Area Covering a triangular piece of chocolate with paper? Surface Area Filling a triangular mold with concrete? Volume