Polygons and Measurement Math 8 Unit 8 Polygons and 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 Strand 4: Concept 4 Measurement 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 128.6 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.

Sum of Interior Angles = Formula: Sum of Interior Angles = 180 (n-2)

Example: Find the sum of the interior angles in the given polygon. a. 14-gon b. 20-gon 180(n-2) 180(n-2) 180(14-2) 180(20-2) 180●12 180●18 Total = 2160  Total = 3240 

Example: Find the measure of each angle in the given regular polygon. a. 16-gon b. 12-gon 180(n-2) 180(n-2) 180(16-2) 180(12-2) 180●14 180●10 Total = 2520  Total = 1800  1800 ÷12 2520 ÷16 150  157.5 

Formula: Sum of Exterior Angles Is always 360

Example: Find the length of each side for the given regular polygon and the perimeter. a.) rectangle, perimeter 24 cm 24 ÷ 4 6 cm b.) pentagon, 55 m 55 ÷ 5 11 m

Example: Find the length of each side for the given regular polygon and the perimeter. c.) nonagon, 8.1 ft 8.1 ÷ 9 0.9 ft d. heptagon, 56 mm 56 ÷ 7 8 mm

Perimeter Evil mathematicians have created formulas to save you time. But, they always change the letters of the formulas to scare you! Any shape’s “perimeter” is the outside of the shape…like a fence around a yard.

Perimeter To calculate the perimeter of any shape, just add up “each” line segment of the “fence”. Triangles have 3 sides…add up each sides length. 8 8 8+8+8=24 The Perimeter is 24 8

Perimeter A square has 4 sides of a fence 12 12 12 12+12+12+12=48 12

Regular Polygons Just add up EACH segment 10 8 sides, each side 10 so 10+10+10+10+10+10+10+10=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 Have Rectangles for faces Named after the shape of their Bases

More Nets

Prisms Fun with by D. Fisher This activity can be done with powerpoint only, but it is much better if the students practice with paper prisms or models made of drinking straws, etc. Besides learning the names of various prisms students can practice “algebraic reasoning.” They can generate sequences and make predictions about patterns. Enjoy this presentation. Send feedback and suggestions to Don Fisher: fisher_d@madera.k12.ca.us by D. Fisher

Vertices (points) 6 Edges (lines) 9 Faces (planes) 5 Triangular Prism Vertices (points) 6 Edges (lines) 9 Faces (planes) 5 The base has 3 sides.

The base has sides. 4 Vertices (points) 8 Edges (lines) 12 Rectangular Prism The base has sides. 4 Vertices (points) 8 Edges (lines) 12 Faces (planes) 6

The base has sides. 5 Vertices (points) 10 Edges (lines) 15 Pentagonal Prism The base has sides. 5 Vertices (points) 10 Edges (lines) 15 Faces (planes) 7

The base has sides. 6 Vertices (points) 12 Edges (lines) 18 Hexagonal Prism The base has sides. 6 Vertices (points) 12 Edges (lines) 18 Faces (planes) 8

The base has sides. 8 Vertices (points) 16 Edges (lines) 24 Octagonal Prism The base has sides. 8 Vertices (points) 16 Edges (lines) 24 Faces (planes) 10

Pyramids Have Triangles for faces Named after the shape of their bases.

Fun with Pyramids By D. Fisher This activity can be done with powerpoint only, but it is much better if the students practice with paper pyramids or models made of drinking straws, etc. Besides learning the names of various pyramids students can practice “algebraic reasoning.” They can generate sequences and make predictions about patterns. Enjoy this presentation. Send feedback and suggestions to Don Fisher: fisher_d@madera.k12.ca.us By D. Fisher

Vertices (points) 4 Edges (lines) 6 Faces (planes) 4 Triangular Pyramid Vertices (points) 4 Edges (lines) 6 Faces (planes) 4 The base has 3 sides.

Vertices (points) The base has sides. 4 5 Edges (lines) 8 Square Pyramid Vertices (points) The base has sides. 4 5 Edges (lines) 8 Faces (planes) 5

Vertices (points) The base has sides. 5 6 Edges (lines) 10 Pentagonal Pyramid Vertices (points) The base has sides. 5 6 Edges (lines) 10 Faces (planes) 6

Vertices (points) The base has sides. 6 7 Edges (lines) 12 Hexagonal Pyramid Vertices (points) The base has sides. 6 7 Edges (lines) 12 Faces (planes) 7

Vertices (points) The base has sides. 8 9 Edges (lines) 16 Octagonal Pyramid Vertices (points) The base has sides. 8 9 Edges (lines) 16 Faces (planes) 9

Name Picture Base Vertices Edges Faces Triangular Pyramid Square Pyramid Pentagonal Pyramid Hexagonal Pyramid Heptagonal Pyramid Octagonal Pyramid 3 4 6 4 4 5 8 5 5 6 10 6 6 7 12 7 Draw it 7 8 14 8 8 9 16 9 No picture n n + 1 2n n + 1 Any Pyramid

Name Picture Base Vertices Edges Faces Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism Heptagonal Prism Octagonal Prism 3 6 9 5 4 8 12 6 5 10 15 7 6 12 18 8 Draw it 7 14 21 9 8 16 24 10 No picture n 2n 3n n + 2 Any Prism

Cylinder Circles for bases Rectangle for side

Points of View View point is looking down on the top of the object. up on the bottom of the object. View point is looking from the right (or left) of the object.

Example 1 : Top Front Top View Side Front View Side View Front Side Bottom View Bottom

Example 2 : Top D q H Front View Left View

Example 3 : Top view Front View Left View

Example 4 Top View Front View Left 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: cm2)

Rectangular Prism SA=2lw +2lh + 2wh l h w

1. Find the surface area. SA=2lw +2lh + 2wh SA=248 + 242 + 282 SA= 112 cm2

SA=2lw +2lh + 2wh SA=266 + 2610 + 2610 SA= 72 + 120 + 120 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 + 2610 + 2610 SA= 72 + 120 + 120 SA= 312 cm2

Cylinder SA =2r2 + 2rh SA = r2 + r2 + hC r C=2r A=hC So A=2rh h

Examples: 1. Find the surface area. SA =2r2 + 2rh SA = 2(5)2 + 2(5)(20) SA = 225 + 2100 SA = 50 + 200 SA = 250 = 785 cm2

SA =2r2 + 2rh SA = 2(9)2 + 2(5)(18) SA = 281 + 290 2. Find the surface area of a cylinder with a height of 5 in and a diameter of 18 in. SA =2r2 + 2rh SA = 2(9)2 + 2(5)(18) SA = 281 + 290 SA = 162 + 180 SA = 342 = 1060.2 in2

Volume Volume: The amount of space inside a 3D shape. Volume is measure in cubic units (ex: cm3)

Rectangular Prism V=LWH V = 842 V = 64 cm3

Cylinder V = 2(5)2(20) V = 22520 V = 2500 SA = 1000 = 3140 cm3 V=r2h V = 2(5)2(20) V = 22520 V = 2500 SA = 1000 = 3140 cm3

Triangular Prism V= ½ LWH V = ½  244910 V = 5, 880 cm3

Surface Area or Volume Covering a Triangular speaker box with carpet? Filling a triangular speaker box with foam? Volume Filling a triangular box with M n M’s?

Surface Area or Volume Painting the outside of a triangular prism? Covering a triangular piece of chocolate with paper? Filling a triangular mold with concrete? Volume