Chapter 12 Notes

12.1 – Exploring Solids

A polyhedron is a solid bounded by polygons 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 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 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. Prism Net View

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

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 LA = Ph Total Area is the sum of ALL the faces. TA = 2B + Ph

Cylinder

Find the LA, SA of this triangular prism. 8 4 3 5

Find the LA, SA of this rectangular prism. 5 3 8

Find the LA and TA of this regular hexagonal prism. If it helps to think like this. 10 4

Find the LA, and TA of this prism.

Find lateral area, surface area Height = 8 cm Radius = 4 cm Height = 2 cm Radius = .25 cm

Find the Unknown Variable. SA = x 2 8 V = 2x x 4

Find the Unknown Variable. SA = 40π cm2 Radius = 4 cm Height = h SA = 100π cm2 Radius = r Height = 4 cm

12.3 – Surface Area of Pyramids and Cones

vertex Lateral edge Altitude (height) Slant height 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.

b l + b l + b l + b l + b l (b + b + b + b + b) l = Pl Lateral Area of a regular pyramid is the area of all the LATERAL faces. Total area is area of bases. TA = B + Pl b l + b l + b l + b l + b l (b + b + b + b + b) l = Pl

Cone

Find Lateral Area, Total Area of regular hexagonal pyramid.

Find Lateral Area, Total Area of regular square pyramid. 13 cm 10 cm

Find lateral area, surface area Find surface area. Units in meters. 6 8 Slant Height = 15 in. Radius = 9 in

Find unknown variable Slant height 8 cm Slant height 8 in Radius = ? x cm Slant height 8 cm Radius = ? TA = 105 cm2 Slant height 8 in Radius = ? TA = 48 π in2

Pyramid height 8 in Slant height 2 cm 20 in 2 cm 12 in

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:

12.4 – Volume of Prisms and Cylinders

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. V = Bh Prism

The volume of an OBLIQUE PRISM is also Bh, remember, it’s h, not lateral edge Altitude. Lateral edge not an altitude. RIGHT PRISM OBLIQUE PRISM.

Find the V of this regular hexagonal prism. 10 4

Find the V of this triangular prism. 8 4 3 5

Find Volume Height = 8 cm Radius = 4 cm Cylinder

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

Cone Slant Height = 15 in. Radius = 9 in 10 cm 13 cm

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

Hexagon is regular, the box is not 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. Units in meters. 8 12

Find Volume Pyramid height 8 in 20 in 12 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

Find SA and V with radius 6 m.

Circumference of great circle 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 Given the measure of the solids, state whether or not they are similar, and if so, what the scale factor is. Surface area of A Surface Area of B Volume of A Volume of B Scale Factor 100 144 125 216 4 9 64 1 8 27

Two similar cylinders have a scale factor of 2:3 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 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.