Concept 52 3-D Shapes

Polyhedron Solid Convex Concave Regular Cylinder Prism Cone Pyramid Sphere

Polyhedron: a solid figure with many plane faces

Solid: a 3-D shape that encloses space but is not made up of all polygon sides.

Convex: all vertices of the solid push outward. Concave: one or more vertices of the solid are pushed inward.

Regular: a polyhedron with all the same regular polygons.

Prism: a polyhedron made up of two parallel bases connected by rectangles. Rectangular prism Triangular Prism Pentagonal Prism

Pyramid:

Cylinder:

Cone:

Sphere:

Determine whether each solid is a polyhedron or solid Determine whether each solid is a polyhedron or solid. Then draw a net for each if possible. 1. 2. 3. 4. Pentagonal Prism Cone Sphere Triangular Prism

Given the net of a solid. Draw the solid and give its name. 5. 6. 7. Triangular Prism Hexagonal Pyramid Cube

Parts of Solids and Cross Section Concept 53 Parts of Solids and Cross Section

Parts of a 3D Shape! Fold on all dotted lines. (fold both directions) Only cut this solid line

Edge: a segment where two faces come together. Base: a polygon Face: a set of polygons that make up the other surfaces of a polyhedron. (lateral faces) Edge: a segment where two faces come together. Vertex: a point where three or more edges come together. Vertex Edge Edge Edge Face Face Vertex Base: Edge Edge Face Edge Vertex Vertex Triangular Pyramid

Then identify the solid Then identify the solid. If it is a polyhedron, name the faces, edges, and vertices. 9. 10. Faces: Edges: Vertices: Pentagons: PWXYX and QRSUV, Quadrilaterals: QVXW, UVXY, USZY, PRSZ, PRXW Faces: Edges: Vertices: Circle S (B) none Point R 𝑃𝑊 , 𝑊𝑋 , 𝑋𝑌 , 𝑌𝑍 , 𝑍𝑃 , 𝑃𝑅 , 𝑊𝑄 , 𝑋𝑉 , 𝑌𝑈 , 𝑍𝑆 , 𝑄𝑉 , 𝑉𝑈 , 𝑈𝑆 , 𝑆𝑅 , 𝑅𝑄 Points: P,W,X,Y, Z,Q,R,S,U,V

Then identify the solid Then identify the solid. If it is a polyhedron, name the faces, edges, and vertices. 11. Faces: Edges: Vertices: Triangles: ABC and DEF, Quadrilaterals: ABED, BCFE, ACFD 𝐴𝐵 , 𝐵𝐶 , 𝐶𝐴 , 𝐴𝐷 , 𝐵𝐸 , 𝐶𝐹 , 𝐷𝐸 , 𝐸𝐹 , 𝐹𝐷 , Points: A, B, C, D, E, F

Cross Section: a surface or shape that is or would be exposed by making a straight cut through something, especially at right angles to an axis. Sketch the cross section from a vertical slice of each figure. 1. 2. 3.

Describe each cross section. 4. 5. 6. Square Triangle Rectangle 7. 8. Oval Rectangle

Concept 54 Euler’s Theorem

There are 11 polyhedrons located around the room at each group of desks. Use each one to fill in a row of the table. If the shape has a name you know write it in the first column, otherwise just write what it is made up of. Ex. (2 triangles and 3 rectangles)

Name or what shapes make it. # of Faces # of Vertices # of Edges 1 2 3 4 5 6 7 8 9 10 11 12

Euler’s Theorem F + V = E + 2

Examples: 8 faces and 18 edges 21 edges and 14 vertices 8+𝑉=18+2 8+𝑉=20 𝑉=12 F+14=21+2 F+14=23 𝐹=9

1 hexagon and 6 triangle faces 12 pentagon faces 8 triangle faces 1 hexagon and 6 triangle faces 12+𝑉=30+2 12∗5=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑑𝑔𝑒𝑠 60=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑑𝑔𝑒𝑠 12+𝑉=32 60 2 =𝑒𝑑𝑔𝑒𝑠 𝑤ℎ𝑒𝑛 𝑝𝑢𝑡 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟 30= 𝑉=20 8+𝑉=12+2 8∗3 2 =𝑒𝑑𝑔𝑒𝑠 8+𝑉=22 12=𝑒𝑑𝑔𝑒𝑠 𝑉=14 7+𝑉=12+2 1∗6 2 + 6∗3 2 =𝑒𝑑𝑔𝑒𝑠 7+𝑉=22 3+9=12=𝑒𝑑𝑔𝑒𝑠 𝑉=15

12 pentagon and 20 hexagon faces 20 triangle faces 12 pentagon and 20 hexagon faces 20+𝑉=30+2 20∗3 2 =𝑒𝑑𝑔𝑒𝑠 20+𝑉=32 30=𝑒𝑑𝑔𝑒𝑠 𝑉=12 12∗5 2 + 20∗6 2 =𝑒𝑑𝑔𝑒𝑠 32+𝑉=90+2 32+𝑉=92 30+60=90=𝑒𝑑𝑔𝑒𝑠 𝑉=60

2 hexagons and 6 rectangles 2∗6 2 + 4∗6 2 =𝑒𝑑𝑔𝑒𝑠 6+12=18=𝑒𝑑𝑔𝑒𝑠 8+𝑉=18+2 8+𝑉=20 𝑉=12

Volume of Prisms Concept 55

Rectangular Prism Triangular Prism Trapezoidal Prism Other Prisms Volume = Base Area ∙ height V = B ∙ h Triangular Prism Trapezoidal Prism Other Prisms

Rectangular Prisms V = B ∙ h V = B ∙ h V = (9 ∙5) ∙ 4 V = (6 ∙11) ∙ 2

Triangular Prism V = B ∙ h V = B ∙ h V = ( 1 2 10∙5) ∙ 14

Trapezoidal Prism V = B ∙ h V = B ∙ h V = ( 1 2 ∙2(4+7)) ∙ 7

Other Prisms V = B ∙ h V = B ∙ h V = ( 1 2 ∙(5.2)(6)(6)) ∙ 7

Find the volume of the right prism. V = B ∙ h V = B ∙ h V = B ∙ h V = (12∙6) ∙8 V = (2∙2) ∙8 V = ( 1 2 ∙4∙3) ∙3 V = 576 𝑚 3 V = 32 𝑐𝑚 3 V = 18 𝑖𝑛 3

Find the missing side length given the Volume, V of each solid. 4. V = 480 cm2 5. V = 120 in2 6. V = 180 cm2 V = B ∙ h V = B ∙ h V = B ∙ h 480 = (12∙𝑥) ∙8 120 = ( 1 2 ∙𝑥∙5) ∙4 180 = ( 1 2 ∙5(𝑥+12)) ∙4 480 = 96x 120 = 10x 180 =10(x +12) 12 in = x 18 = x + 12 5 cm = x 6 cm = x

Volume of Pyramids Concept 56

Volume of Pyramids Square Pyramid Triangular Pyramid Other Pyramids

Volume of Pyramids Volume = 1 3 ∙ Base Area ∙ height V = 1 3 ∙ B ∙ h

Square Pyramids V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h V = 1 3 ∙(7∙3)∙ 8 1. 2. V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h V = 1 3 ∙(7∙3)∙ 8 V = 1 3 ∙(10∙10)∙ 7 V = 56 𝑐𝑚 3 V = 233.3 𝑘𝑚 3

Triangular Pyramids V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h 3. 4. V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h V = 1 3 ∙( 1 2 ∙4∙6)∙ 7 V = 1 3 ∙( 1 2 ∙3∙4)∙5 V = 28 𝑐𝑚 3 V = 10 𝑐𝑚 3

Other Pyramids V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h 5. 6. V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h V = 1 3 ∙( 1 2 ∙4.1∙5∙6)∙12 V = 1 3 ∙(5∙4)∙6 V = 246 𝑓𝑡 3 V = 40 𝑐𝑚 3

Find the volume of each pyramid Find the volume of each pyramid. Round to the nearest tenth if necessary. 𝑎 2 + 𝑏 2 = 𝑐 2 6 2 + 𝑥 2 = 10 2 V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h 36+ 𝑥 2 =100 V = 1 3 ∙(3∙3)∙7 V = 1 3 ∙(12∙8)∙10 𝑥 2 =64 𝑥=8 V = 21 𝑖𝑛 3 V = 320 𝑓𝑡 3 V = 1 3 ∙ B ∙ h V = 1 3 ∙( 1 2 ∙6∙8)∙15 V = 120 𝑓𝑡 3

V = 1 3 ∙ B ∙ h V = 1 3 ∙( 1 2 ∙6∙8)∙15 V = 120 𝑓𝑡 3

Volume of Cylinders, cones, and spheres Concepts 57 - 59

Find the volume of each. 1. 2. 3.

4. 5. 6.

7. hemisphere: area of 8. 9. great circle ≈ 4π ft2