Have Out: Bellwork: 7/11/12 Homework, red pen, pencil, gradesheet Find the Area 2 h2 = 122 + 92 (Pythagorean. Thm.) 15 ft h2 = 144 + 81 h 10 ft h = 15 ft P = 2 + 15 + 10 + 24 + 15 24 ft = 66 ft.
A = + = (10 15) + ( 12 9) = 150 + 54 = 204 ft2 1 Picture Equation Formulas 10 ft 3 Simplify 4 Solve 24 ft A = + 10 12 15 9 1 = (10 15) + ( 12 9) 2 = 150 + 54 = 204 ft2
Surface Area: Add the area of every side
( ) SA = + 2 + = (½ • 10 • 12) + 2(½ • 18 • 8) + (½ • 9 • 8) Net: 3-D figure unfolded (Helps you see all the sides) 12 SA = + 2 + 10 ( ) 18 8 9 = (½ • 10 • 12) + 2(½ • 18 • 8) + (½ • 9 • 8) = (60) + 2(72) + (36) = 60 + 144 + 36 = 240 u2
Add to your notes... Total Surface Area: The sum of the areas of EACH of the faces of a polyhedron. We can set up TSA problems just like we did the area subproblems! One equation…four steps: 1. Picture Equation 2. Formulas 3. Simplify 4. Solve & Answer with correct units. What type of units would be correct for Total Surface Area? units2
How many faces does the Rectangular Pyramid on the resource page have? SV-101 How many faces does the Rectangular Pyramid on the resource page have? 5 15 TSA = + 4 ( ) 20 1. Picture Equation = (15 • 15) + 4(½ • 20 • 15) 2. Formulas = (225) + 4(150) 3. Simplify = 225 + 600 4. Solve & Answer with correct units. = 825 u2
Add to your notes: PYRAMID Base: Polygon on the bottom Rectangular Pyramid PYRAMID Base: Polygon on the bottom base The shape of the base gives the figure it’s name lateral faces Lateral faces: Triangles that connect the base to one point at the top. (vertex) The lateral faces are not always the same size base Hexagonal Pyramid
12’ 10’
Add to your notes: A PRISM is: base 2 congruent (same size and shape) parallel bases that are polygons height 3) lateral faces (faces on the sides) that are parallelograms formed by connecting the corresponding vertices of the 2 bases. base lateral faces Lateral faces may also be rectangles, rhombi, or squares.
rectangular, square, or parallelogram prism pentagonal prism On your paper, shade the figure that is the base for each of the following solids. Then name the solid using the name of its polygonal base and either prism or pyramid. triangular prism rectangular, square, or parallelogram prism pentagonal prism hexagonal prism triangular pyramid octagonal prism rectangular, square, or trapezoidal pyramid pentagonal pyramid Rectangular or parallelogram prism triangular prism pentagonal prism hexagonal pyramid
( ) V= • 20 = (84)(20) = (32)(20) = 640 u3 TSA = 2 + 2 + 2 10 8 TSA = 2 + 2 + 2 ( ) 4 20 = 2(4 • 8) + 2(4 • 20) + 2(20 • 8) = 2(32) + 2(80) + 2(160) = 64 + 160 + 320 = 544 u2
V= • 12 V= • 15 = (52)(12) = (½ 12 • 5)(15) = 624 cm3 = (30)(15) 1. 2. 12 in 5 in 15 in 12 cm 13 in Area of Octagon =52 cm2 12 V= • 12 V= • 15 5 = (52)(12) = (½ 12 • 5)(15) = 624 cm3 = (30)(15) = 450 in3
Polyhedra Add to your notes... Polyhedron: SV-83 Polyhedra Add to your notes... Polyhedron: A 3-dimensional object, formed by polygonal regions, that has no holes in it. Plural: polyhedra face: A polygonal region of the polyhedron. edge: A line segment where two faces meet. vertex: A point where 3 or more sides of faces meet. Plural: vertices faces vertices edges
These are NOT polyhedra: SV-83 These are polyhedra: These are NOT polyhedra:
SV-84 Classify the following as a polyhedron or not a polyhedron. Write YES or NO. If no, explain why not. YES YES NO The face has a curve, which is not a polygon. NO It is only 2-dimensional. YES NO It is only 2-dimensional. The face has a curve, which is not a polygon. It is only 2-dimensional. NO NO YES
Be familiar with these names. SV-84 Polyhedra are classified by the number of faces they have. Here are some of their names: 4 faces tetrahedron 9 faces nonahedron 5 faces pentahedron 10 faces decahedron 6 faces hexahedron 11 faces undecahedron 7 faces heptahedron 12 faces dodecahedron 8 faces octahedron 20 faces icosahedron Be familiar with these names.
Then, use the information you found to answer SV-79 Complete your resource page by counting the total number of vertices, edges, and faces for each polyhedron. Then, use the information you found to answer SV-80
4 6 4 8 8 12 6 14 6 12 8 14 12 18 8 20 10 15 7 17 20 30 12 32
A) For each row, calculate VR + F. See resource page. SV-80 Let VR = number of vertices, E = number of edges, and F = number of faces. In 1736, the great Swiss mathematician Ledonhard Euler found a relationship among VR, E, and F. A) For each row, calculate VR + F. See resource page. Write an equation relating VR, F, and E. VR + F = E + 2 Ledonhard Euler 1707-1783
Yes it is possible. Shorten the length of any one edge. SV-81 Is it possible to make a tetrahedron with non-equilateral faces? If not, explain why not. If so, draw a sketch. Yes it is possible. Shorten the length of any one edge. Possible examples:
How many vertices does it have? SV-82 How many edges does the solid have? (Don’t forget the ones you can’t see.) There are 9 edges. How many vertices does it have? There are 6 vertices.