Measurement – 3D

1. Measurement – 3D Right Prisms & Cylinders, Right Pyramids & Cones, Platonic Solids, Composite Figures

2. Table of Contents • 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!

3. Right Solids • 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.

4. Cylinder • Two congruent circles connected by one rectangle. • Area of the base is πr2. • Volume = πr2*h. • Lateral Surface Area is one rectangle. The area of the rectangle is circumference of the base times height. LSA =2πr*h. • SA = 2(πr2)+ 2πr*h. h 2πr πr2

5. Cylinder What is missing? • Find volume given h=10 and r=5. Round to.001. • Find r given h=13 and V= 52π. • Find SA given h=3 and V=12π. • Find SA given that d=14 and h=10. • Find h given that d=8 and V=220. Round to the 10th.

6. Interior Angle of a Regular Polygon The long way which creates understanding… • 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. 135⁰ 45⁰ 67.5⁰ 67.5⁰ 360⁰ 180⁰-45⁰ = 45⁰ = 67.5⁰ 8 2

7. Interior Angle of a Regular Polygon The shortcut! 180⁰(n-2) 135⁰ n 180⁰(8-2) = 135⁰ 8

8. Area of a Regular Polygon 14 • 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. 12 a s A=nΔ=n(½sa) A=6(½*14*12)=504 A=n(½sa)=½(ns)a=½Pa

9. Right Prism • 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). s a n(s*h) nΔ = ½ Pa h s*h

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

11. Pyramids & Cones V= 1/3πr2*h LA= 1/3(2πr)*l • 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. l h πr2 V= 1/3s2*h LA= 1/2 P*l l h s2

12. Pyramids • 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 V=1/3(½Pa)*h l2=h2 + a2

13. Coming Soon – Cones!

14. Platonic Solids

15. Why are there only five? • What is the upper limit of the sum of the angles of a vertex of a polyhedra? • Find the interior angles of the regular: triangle, square, pentagon, hexagon, septagon, octagon, & nonagon. • The sum must be less than 360⁰ or the “corner” will be flat. • The interior angle of a regular triangle is 60⁰, square is 90⁰, pentagon is 108⁰, hexagon is 120⁰, septagon is 128.57⁰, octagon is 135⁰, & nonagon is 140⁰. 60⁰ 90⁰ 108⁰ 120⁰ 128.57⁰ 135⁰

16. Why are there only five? Cont. • What is happening? • What is the minimum number of faces that will meet at the vertex of a polyhedra? • 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?) • As the number of sides increases, the interior angle gets closer to 180⁰. • There must be at 3 faces or the figure will not be 3D. • Triangle, Square, & Pentagon. 90⁰*3<360 ⁰ 108⁰*3<360 ⁰ 60⁰*3<360 ⁰ 60⁰*4<360 ⁰ 60⁰*5<360 ⁰

17. Why are there only five? Cont. • 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.

18. Faces and Vertices and Edges, Oh My! • What is the number of faces, vertices, & edges of a pentagonal prism? • What is the number of faces, vertices, & edges of a triangular prism? • What is the number of faces, vertices, & edges of a pentagonal pyramid? • What is the number of faces, vertices, & edges of a tetrahedron? • What are the patterns? • In a pentagonal prism, there are 7 faces, 10 vertices & 15 edges. • In a triangular prism, there are 5 faces, 6 vertices & 9 edges. • In a pentagonal pyramid, there are 6 faces, 6 vertices & 10 edges. • In a tetrahedron, there are 4 faces, 4 vertices & 6 edges.

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