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