Right Prisms & Cylinders, Right Pyramids & Cones, Platonic Solids, Composite Figures

Right Solids – Cylinders – Interior Angle of a Polygon – Area of a Polygon – Right Prisms Pyramids & Cones – Pyramids – Cones Platonic Solids – Why Only 5? – Tetrahedron – Cube (Hexahedron) – Octahedron – Dodecahedron – Icosahedron Composite Figures Faces & Vertices & Edges, Oh My!

Two congruent ends connected by rectangle(s). One end is the “Base”. The “height” connects the two ends. Volume is Base times height. Surface Area is 2 times Base + the rectangles connecting the ends.

Two congruent circles connected by one rectangle. Area of the base is πr 2. Volume = πr 2 * h. Lateral Surface Area is one rectangle. The area of the rectangle is circumference of the base times height. LSA = 2πr * h. SA = 2(πr 2 ) + 2πr * h. 2πr2πr πr2πr2 h

1)Find volume given h=10 and r=5. Round to )Find r given h=13 and V= 52π. 3)Find SA given h=3 and V=12π. 4)Find SA given that d=14 and h=10. 5)Find h given that d=8 and V=220. Round to the 10 th.

Regular polygons are made of congruent isosceles triangles. Around the center of the polygon there is 360⁰. Divide 360⁰ by the number of sides and you have the apex angle of the isosceles triangle. Subtract the apex angle from 180⁰, divide by 2 and you have the base angles of the isosceles triangle. The interior angle is twice the base angle. 360⁰ 8 = 45⁰ 180⁰-45⁰ 2 = 67.5⁰ 45⁰ 67.5⁰ 135⁰ The long way which creates understanding…

180⁰(n-2) n = 135⁰ 135⁰ The shortcut! 180⁰(8-2) 8

Regular polygons are made of congruent isosceles triangles. The apothem of the polygon is the altitude of the isosceles triangle. The side of the polygon is the base of the isosceles triangle. Area of the polygon is n times the area of the isosceles triangle. Area of the polygon is also ½ the perimeter times the apothem. A=nΔ=n(½sa) A=6(½*14*12)= A=n(½sa)=½(ns)a=½Pa a s

Two congruent n-gons connected by n rectangles. If the polygons are regular then Area of the base is nΔ=n(½ s * a)= ½ Pa. Volume = (nΔ) * h=(½Pa) * h. Lateral Surface Area is n rectangles. The area of one rectangle is side of the polygon times height. LSA = n(s * h). SA = 2(½Pa) + n(s * h). n(s * h) nΔ = ½ Pa hs*hs*h s a

1)Find V & SA of a triangular prism given h (of prism)=5, s=12, a=6√3. 2)Find s of a hexagonal prism given h=13 and V= 1950√3. 3)Find P of a hexagonal prism given h=10, a=7√3 and V= 2940√3. 4)Find SA of a octagonal prism given that h=14 and s=10. 5)Find V & SA of a pentagonal prism given h=10 and s=14. Round to.001. All polygons are regular.

One end connected to a point directly over the center of the end. The end is the “Base”. The “height” connects the center to the point. Volume is 1/3 Base times height. Lateral Area connects the base to the point. Surface Area is Base + lateral surface area. πr2πr2 h V= 1 / 3 πr 2 * h h s2s2 V= 1 / 3 s 2 * h l LA= 1 / 3 (2πr) * l l LA= 1 / 2 P * l

The base can be any polygon. The lateral area, “sides”, are triangles. The lateral area is the sum of the areas of all the triangles. The side of the polygon is the base of the triangle. The altitude of the triangle is the slant height of the pyramid. Surface Area = Base + Lateral Surface Area. If the polygon is regular than you can use the surface area formula on the reference sheet. The height, apothem, and slant height form a right triangle. Volume = 1/3Base*height LA= n(½ s * l )=½(ns) * l =½P * l SA=½P * l + B l 2 =h 2 + a 2 V=1/3(½Pa)*h

1)What is the upper limit of the sum of the angles of a vertex of a polyhedra? 2)Find the interior angles of the regular: triangle, square, pentagon, hexagon, septagon, octagon, & nonagon. 1)The sum must be less than 360⁰ or the “corner” will be flat. 2)The interior angle of a regular triangle is 60⁰, square is 90⁰, pentagon is 108⁰, hexagon is 120⁰, septagon is ⁰, octagon is 135⁰, & nonagon is 140⁰. 60⁰ 90⁰ 108⁰ 120⁰ ⁰ 135⁰

3)What is happening? 4)What is the minimum number of faces that will meet at the vertex of a polyhedra? 5)What congruent regular polygon can meet at a vertex? (i.e. What are the regular polygons that can be the faces of a regular polyhedra?) 3)As the number of sides increases, the interior angle gets closer to 180⁰. 4)There must be at 3 faces or the figure will not be 3D. 5)Triangle, Square, & Pentagon. 60⁰*3<360 ⁰ 90⁰*3<360 ⁰ 108⁰*3<360 ⁰ 60⁰*5<360 ⁰ 60⁰*4<360 ⁰

Describe the five platonic solids by face and number of polygons that meet at a vertex. Justify why each platonic solid is possible using the previous reasoning.

1)What is the number of faces, vertices, & edges of a pentagonal prism? 2)What is the number of faces, vertices, & edges of a triangular prism? 3)What is the number of faces, vertices, & edges of a pentagonal pyramid? 4)What is the number of faces, vertices, & edges of a tetrahedron? 5)What are the patterns? 1)In a pentagonal prism, there are 7 faces, 10 vertices & 15 edges. 2)In a triangular prism, there are 5 faces, 6 vertices & 9 edges. 3)In a pentagonal pyramid, there are 6 faces, 6 vertices & 10 edges. 4)In a tetrahedron, there are 4 faces, 4 vertices & 6 edges.

What can you conclude about the sum of the exterior angles of a polygon?