11.1 Space Figures and Cross Sections

Polyhedron: A 3-dimensional figure whose surfaces are polygons.

Faces: Each polygon surface.

Edge: The segment formed by the intersection of the faces.

Vertex: The point where edges intersect.

Ex: How many vertices, edges, and faces? List them.

Vertices: 6 (R,S,T,U,V,W)

Edges: 9 (RS, RV, SW, VW, UV,TW, UT, RU, ST)

Faces: 5 (ΔRUV, ΔSTW, STUR, SRVW, UTWV)

F + 20 = 30 + 2 20 + 12 = E + 2

F = 12 30 = E

Net: A 2-dimensional drawing that you can fold and make a 3-D figure.

Draw a net for the pyramid.

Cross Section: The intersection of a solid and a plane. (A slice of the solid)

Ex: What are the cross sections of the cones?

Triangle Circle

11.2 Surface Area of Prisms and Cylinders

Prism: A polyhedron with two congruent and parallel faces (bases)

-Name prisms using the shape of the base.

-Lateral Faces: The non parallel faces

Lateral Area: The sum of the areas of all the lateral faces.

Surface Area: The sum of the areas of all the faces (lateral and bases).

Find the surface area.

1. Lateral Area: (LA = ph)

= (4+4+3+3) (5) = 70

2. Area of base: 4(3) = 12

3. Surface Area: LA + 2(Area of Base)

= 70 + 2(12) = 94 cm²

1. Find 3rd side of the triangular base:

3² + 4² = x²

X = 5

2. Lateral Area (LA = ph)

LA = (3 + 4 + 5)(6) = 72

3. Area of base: ½(3)(4) = 6

4. Surface Area: LA + 2(Area of Base)

= 72 + 2(6) = 84 cm²

Find the surface area.

LA = 2πrh = 2π(2)(10) = 40π

Area of base: 2²π = 4π

Surface Area: LA + 2Area of Base

= 40π + 2(4π) = 48π m²

LA = 2π(5)(30) = 75π

Area of base: 5² π = 25π

SA = 75π + 2(25π) = 125π cm²

11.3 Surface Area of Pyramids and Cones

Height: The altitude of the whole pyramid.

Slant height: Height of the lateral face

Find the surface area.

LA= ½ pl

= ½(52)(8) = 208

Area of base = 169

SA = LA + Area of base

= 208 + 169 = 377 m²

Area of base = 100

Slant height: 5² + 12² = c²

169 = c²

C = 13

LA = ½(40)(13) = 260

SA = 260 + 100 = 360 m²

Find the surface area. Leave in terms of π.

LA = πrl

= 6(12)π = 72π

Area of base = 6²π = 36π

SA = LA + Area of base

= 72π + 36π = 108π in²

Find the surface area. Round to the nearest tenth.

Slant height = 8² + 14² = c²

260 = c²

C = 16.1

LA = 8(16.1)π = 404.6

Area of the base: 8²π = 201.1

SA = 404.6 + 201.1 = 605.7 in²

11.4 Volumes of Prisms and Cylinders

Volume: The space that a figure occupies.

Ex: Find the volume of the prism.

V = 5(3)(4) = 60 ft³ V = ½(10)(6)(5) = 150 m³

Ex. Find the volume. Leave in terms of π.

V = 2²π(10) = 40π m³

11.5 Volumes of Pyramids and Cones

Ex: Find the volume.

V = 1/3(10)(10)(9) = 300 in³

Height: x² + 5² = 12²

x² = 119

x = 10.9

V = 1/3(10)(10)(10.9) = 363.3 m³

Ex. Find the volume.

V= 1/3(64π)(14) = 298.7π in³

Height: h² + 6²= 12²

h = 10.4

V = 1/3(36π)(10.4) = 124.8π in³

11.6 Surface Area and Volumes of Spheres

Ex: Find the surface area.

SA = 4(5²)π = 100π m² SA = 4(6²)π = 144π cm²

Ex: Find the surface area given the circumference.

A grapefruit with C = 14 cm

π d = 14

d = 14/π = 4.5 (r = 2.25)

SA = 4(2.25²)π = 63.6

Ex. Find the volume.

V = 4/3(5³)π = 4/3(25π) = 100π/3 V = 4/3(6³)π = 4/3(216π) = 288π

Ex. A sphere has a volume of 900 in³. What is the surface area?

4/3 r³π = 900

r³π = 675

r³ = 214.9

r = 6.0

SA = 4(6²)π = 144π in²

11.7 Areas and Volumes of Similar Solids

Similar solids: Solids with the same shape and proportional dimensions.

Ex:

Yes 6/5

Ex. Compare the first figure to the second.

Scale Factor: ½

Area scale factor: ¼

Volume scale factor: 1/8

Ex. Each pair of figures is similar. Find the scale factor of the smaller figure to the bigger figure.

250π/432π

Simplify to get 125/216

Cube root to get 5/6

18/32

Simplify to 9/16

Square root to get ¾