chapter 12 notes. 12.1 – exploring solids a polyhedron is a solid bounded by polygons. sides are...

Chapter 12 Notes

12.1 – Exploring Solids

A polyhedron is a solid bounded by polygons. Sides are faces, edges are line segments connecting faces, and vertices are points where the edges meet. The plural of polyhedron is polyhedra or polyhedrons.

A polyhedron is regular if all the faces are congruent regular polygons.

A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely on the polyhedron (like with polygons)

There are 5 regular polyhedra, called Platonic solids (they’re just friends). They are a (look at page 721):

tetrahedron (4 triangular faces) a cube (6 square faces) octahedron (8 triangular faces) dodecahedron (12 pentagonal faces) icosahedron (20 triangular faces)

http://personal.maths.surrey.ac.uk/st/H.Bruin/image/PlatonicSolids.gif

Is it a polyhedron? If so, count the faces, edges, and vertices. Also say whether or not it is convex.

No

Euler’s Theorem:

(Like FAVE two, or cube method)

Edges = Sides/2

Find vertices of polyhedra made up of 8 trapezoids, 2 squares, 4 rectangles.

Find vertices of polyhedra made up of 2 hexagons, 6 squares.

The intersection of a plane crossing a solid is called a cross section. Sometimes you see it in bio, when they show you the inside of a tree, the circle you get when you slice a tree is called the cross section of a tree.

12.2 – Surface Area of Prisms and Cylinders

Terms

P Perimeter of one base

B Area of base

b base (side)

h Height (relating to altitude)

l Slant Height

TA Total Area (Also SA for surface Area)

LA Lateral Area

V Volume

Prism Net View

A prism has two parallel bases.

Altitude is segment perpendicular to the parallel planes, also referred to as “Height”.

Lateral faces are faces that are not the bases. The parallel segments joining them are lateral edges.

If the lateral faces are rectangles, it is called a RIGHT PRISM. If they are not, they are called an OBLIQUE PRISM.

RIGHT PRISM OBLIQUE PRISM.

Altitude.

Lateral edge not an altitude.

Lateral Area of a right prism is the sum of area of all the LATERAL faces.

= bh + bh + bh + bh= (b + b + b + b)h

= PhLA

Total Area is the sum of ALL the faces. = 2B + PhTA

LABrhrSA

PhrhLA

222

22

Cylinder

3

5

Find the LA, SA of this triangular prism.

84

LA =

SA =

3

5

Find the LA, SA of this rectangular prism.

8

LA =

SA =

10

4

Find the LA and TA of this regular hexagonal prism.

If it helps to think like this.

Find the LA, and TA of this prism.

6 in10 in

8 in

24 in

30 in

Height = 8 cm

Radius = 4 cm

Find lateral area, surface area

Height = 2 cm

Radius = .25 cm

2

x

Find the Unknown Variable.

8

SA =2192units

x

2x

4

V =372units

SA = 40π cm2

Radius = 4 cm

Height = h

Find the Unknown Variable.

SA = 100π cm2

Radius = r

Height = 4 cm

12.3 – Surface Area of Pyramids and Cones

vertex

Altitude

(height)Slant height

Lateral edge

Lateral Face (yellow)

Base (light blue)

A regular pyramid has a regular polygon for a base and its height meets the base at its center.

Lateral Area of a regular pyramid is the area of all the LATERAL faces.

2

1b l +

2

1b l +

2

1b l+ 2

1b l +

2

1b l

2

1l(b + b + b + b + b) = Pl

2

1

Total area is area of bases. TA = B + Pl

2

1

LABrlrSA

PlrlLA

2

2

1

Cone

Find Lateral Area, Total Area of regular hexagonal pyramid.

10 in

16 in

10 cm

13 cm

Find Lateral Area, Total Area of regular square pyramid.

6

8

Slant Height = 15 in.

Radius = 9 in

Find lateral area, surface area

Find surface area. Units in meters.

Slant height 8 in

Radius = ?

TA = 48 π in2

Find unknown variable

x cm

8 cm

Slant height 8 cm

Radius = ?

TA = 105 cm2

Pyramid height 8 in

12 in

20 in

2 cm

Slant height

2 cm

Look at some cross sections

12.1 – 12.3 – More Practice, Getting Ready for next week

Find the length of the unknown side

Find the area of the figures below, all shapes regular

Find the area of the shaded part

To save time, formulas for 12.4 are as follows:

hrV

BhV

2

cylinderofVolume

PrismofVolume

12.4 – Volume of Prisms and Cylinders

Prism

Altitude is segment perpendicular to the parallel planes, also referred to as “Height”.

Volume of a right prism equals the area of the base times the height of the prism.

= BhV

The volume of an OBLIQUE PRISM is also Bh, remember, it’s h, not lateral edge

RIGHT PRISM OBLIQUE PRISM.

Altitude.

Lateral edge not an altitude.

10

4

Find the V of this regular hexagonal prism.

3

5

Find the V of this triangular prism.

84

V = 8)4)(3(2

1

348 units

BhhrV 2Cylinder

Find Volume

Height = 8 cm

Radius = 4 cm

Circumference of a cylinder is 12π, and the height is 10, find the volume.

Find the unknown variable.

What is the volume of the solid below?

What is the volume of the solid below? Prism below is a cube.

12.5 – Volume of Pyramids and Cones

BhV3

1

:is pyramid a of volumeThe

10 cm

13 cm

BhhrV3

1

3

1 2

Cone

Slant Height = 15 in.

Radius = 9 in

Circumference of a cone is 12π, and the slant height is 10, find the volume.

3396

is volume theandin 8 islength side

theif base trianglelequilateraan

withpyramid a ofheight theFind

in

8

12

Find volume. Units in meters.

Hexagon is regular, the box is not. Hexagon radius 4 units, height is 6 units Finding the volume of box with hexagonal hole drilled in it.

Find Volume

Pyramid height 8 in

12 in

20 in

12.6 – Surface Area and Volume of Spheres

A sphere with center O and radius r is the set of all points in SPACE with distance r from point O.

Great Circle: A plane that contains the center of a circle.

Hemisphere: Half a sphere.

Chord: Segment whose endpoints are on the sphere

Diameter: Segment through center of the sphere

3

2

3

4

4

rV

rSA

Find SA and V with radius 6 m.

Radius of Sphere

Circumference of great circle

Surface Area of Sphere

Volume of sphere

3 m

4π in2

6π cm

9π ft3

2

Find the area of the cross section between the sphere and the plane.

Radius 4 in. Cylinder height 10 in. Find area, volume

Find volume, side length of cube is 3 in.

12.7 – Similar Solids

Find the total area and volume of a cube with side lengths:

Area Volume

1

2

5

10

Two shapes are similar if all the the sides have the same scale factor.

If the scale factor of two similar solid is a:b, then

The ratio of the corresponding perimeters is a:b

The ratio of the base areas, lateral areas, and total areas is a2:b2

The ratio of the volumes is a3:b3

Surface area of A

Surface Area of B

Volume of A

Volume of B

Scale Factor

100 144 125 216

4 9 64 125

1 4 8 27

Given the measure of the solids, state whether or not they are similar, and if so, what the scale factor is.

Two similar cylinders have a scale factor of 2:3. If the volume of the smaller cylinder is 16π units3 and the surface area is 16π units2, then what is the surface area and volume of the bigger cylinder?

Two similar hexagonal prisms have a scale factor of 3:4. The larger hexagon has side length 4 in and height 9 in. Find the surface area and volume of the smaller prism using ratios.