Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right prisms, regular pyramids, cones and cylinders.

What are SOLIDS? We start with points to get lines…and then lines to form different types of polygons and circles Then… If we take these different figures and put them together, we get SOLIDS. YOU MUST SEE THE WHOLE, BUT ALSO THE SUM OF ITS PARTS!!

Formulas from Ch. 11 Area of.. Perimeter for any polygon… Square Rectangle Triangle Trap Rhombus Regular polygon Circle Perimeter for any polygon… Length of sides added together Perimeter for a circle (circumference) ∏2r

12-1: Prisms Objectives Learn and apply the area and volume formula for a prism.

Anatomy of a Prism Base Lateral edge Altitude Lateral Face Base Congruent polygons lying in parallel planes Base Lateral edge Altitude Lateral Face Base Base Edge Altitude is the height of the prism, it is perpendicular to both bases

The lateral faces of a right prism are always rectangles Oblique vs. Right The lateral edges of a right prism are perpendicular to the bases. The lateral faces of a right prism are always rectangles

Names of Prisms The name of a prism comes from its base. Rectangular Oblique Prism Triangular Right Prism

Lateral Area **sum of the areas of each lateral face** p = perimeter of the base h = height of the lateral edge THIS FORMULA APPLIES TO ANY RIGHT PRISM L.A. = ph

Lateral Area of a Hexagonal Prism Find the lateral area of the regular hexagonal prism. The bases are regular hexagons. So the perimeter of one base is 6(5) or 30 centimeters. L.A. = ph Lateral area of a prism P = 30, h = 12 Multiply. Answer: The lateral area is 360 square centimeters.

Total Area **THINK GIFT WRAPPING*** Total Area = L.A. + 2B L.A. – Lateral Area B – Area of base (area of base depends on the shape of the poly)

Find the surface area of the triangular prism. A. 320 units2 B. 512 units2 C. 368 units2 D. 416 units2 A B C D

Volume **think inside the box** B = the area of the base h = height of the lateral edge Volume is cubic units i.e. units3, cm3, in3 THIS FORMULA APPLIES TO ANY RIGHT PRISM V = B x h

Find the volume of the triangular prism. V Bh Volume of a prism 1500 Simplify. Answer: The volume of the prism is 1500 cubic centimeters.

WHITEBOARDS Classroom ex. 1-5 , 7, 8

White Board Practice The base of a triangular prism is a right triangle with legs of 3 cm and 4cm. The height of the prism is 20 cm. Find the lateral area, total area, and volume of the prism L.A = 240cm2 Total area = 252cm2 V = 120cm3

h = 15 White Board Practice A right triangular prism has base edges of 5, 12, and 13. It has a volume of 450. Find the height of the prism. h = 15

Set up HW Breakdown the diagram on 1 - 6

Special Prism – THE CUBE What is special about the edges of a cube? EVERY EDGE IS THE SAME LENGTH!! Q#1: V = 512, find edge and TA Q#2: TA = 216 , find V. Edge length = TA = V = x

12-2:Pyramids Objectives Learn and apply the area and volume formula for a pyramid.

Each lateral face is a triangle Anatomy of a Pyramid Vertex Lateral Face Each lateral face is a triangle Lateral edge Altitude Base Slant Height Apothem (length = ½ length of square side) Base Edge Altitude is the height of the pyramid, from the vertex perpendicular to the base

Regular Pyramids Base is a regular polygon (ex. Square) Each face is a congruent isosceles triangle pg. 482: 3rd paragraph Not regular Regular

Rotate the pyramid….. this is hard (looking at a lateral face) The slant height is the height of each lateral face The slant height is the leg of one right triangle and the hypot of another one. Lateral edge Partners: Look at your labels for a pyramid…how would you label this triangle using those labels? l Slant Height Base Edge

l l l s s Height h Lateral edge Base Edge Slant Height Slant Height Apothem of regular poly Base Edge

Find the missing pieces Lateral edge h l a s s Find the missing pieces h = 6√3 , s = 12 Lateral edge = 6, s = 4 l = 4 , a = 3

Regular Square Pyramid Names of Pyramids The name of a pyramid comes from its base. Y D C B A Triangular Oblique Pyramid Regular Square Pyramid Y-ABCD

Lateral Area **sum of the areas of each lateral face** Method 1: Find the area of one lateral face and multiply by # of faces Method 2: p = perimeter of base l = slant height l s s

Total Area = L.A. + B L.A. – Lateral Area B – Area of base (area of base depends on the shape of the regular poly) Video

Volume Think about the beach…. h h B B It would take 3 pyramids of sand to fill a bucket that was in the shape of a prism with the same base and height h h B B

White Board Practice A regular square pyramid had base edge 6m and lateral edge 5m. a) Find the length of a slant height b) Find the Lateral Area c) Find the Base Area d) Find the Total Area e) Find the Length of the altitude f) Find the Volume 4m 48m2 36m2 5m 84m2 √7m 12√7m3 6m

Whiteboard P. 484 Classroom ex 2 - 7

12-3: Cylinders and Cones Objectives Find the area and volume for a cylinder and a cone.

WARM – UP in NOTES Read pg. 490 & Top of 491 List the similarities between a prism and a cylinder How do you think the formulas (L.A – T.A – V) are going to compare? Do the same for pyramids compared to cones..

Why is this a right cylinder? A cylinder is a prism with a circular base. Why is this a right cylinder? h h r Base Base

Total Area **THINK GIFT WRAPPING*** Right Prism Lateral Area L.A. = ph Total Area TA = L.A + 2B Right Cylinder Lateral Area L.A. = (2Пr) h Total Area TA = L.A. + 2 (Пr2)

Find the total area of the cylinder. The radius of the base and the height of the cylinder are given. Substitute these values in the formula to find the surface area. Total area of a cylinder =896∏

How do we find the volume of a right cylinder? Partners: How do you find the volume of a right prism? V = Bh V = (Пr2) h How do we find the volume of a right cylinder? B = Пr2 h = height of the cylinder Volume is cubic units i.e. units3, cm3, in3 h r B

Whiteboards - Cylinders L. A. TA V 5 10 √7 3 12 1 2 3

whiteboards The volume of a cylinder is 64∏. If r = h, find r. 4

Cone l l A cone is a pyramid with a circular base. Why is this a right cone? l Base h l h r Base

Total Area **THINK GIFT WRAPPING*** Regular Pyramid Lateral Area L.A. = ½ pl Total Area TA = L.A + B Right Cone Lateral Area L.A. = ½ (2Пr) l = П(r)(l) Total Area TA = L.A. + (Пr2) video

How do we find the volume of a right cone? Partners: How do you find the volume of a right pyramid? V = 1/3 Bh V = 1/3 (Пr2)h How do we find the volume of a right cone? B = Пr2 h = height of the cone Volume is cubic units i.e. units3, cm3, in3 h r B

White Board Practice Find the lateral area, total area and volume of a cone with height and radius 3 cm. LA = 18 cm2 TA = 27 cm2 V = 9  cm3 ? 3√3cm 3 Base

White Board Practice Find the lateral area, total area and volume of a cone with slant height 13 cm and radius 12 cm. LA = 156 cm2 TA = 300 cm2 V = 240  cm3 Base

12-4: Spheres Objectives Determine the area and volume of a sphere.

Sphere A sphere is the locus (set) of points in space equidistant from a given point. r

Area of a Sphere **how much paint do I need** Partners: The radius of a baseball is 2in,what is the area of the baseball? r 16 П in2

Volume of a Sphere **how much air do I need** V = 4/3 Пr3 Partners: The diameter of a basketball is 12in,what is the volume of the basketball? r 288 П in3

What is the volume of a sphere if the area is 324∏? What is the common element we must find in both formulas? r =radius What is the volume of a sphere if the area is 324∏? r = 9 ; V = 972 ∏

White Board Practice

144cm2 White Board Practice Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm. 15 9 144cm2

If the volume of a hemisphere is 18П, what is the area of the total sphere? Hint: find radius first 36П

White Board Practice Betty made two wax candles, one in the shape of a sphere with radius 5cm and another in the shape of a cylinder with radius 5cm and height 6 cm. Which candle required more wax? Volume of Sphere = 166 2/3  cm3 Volume of Cylinder = 150  cm3

12-5: Areas and Volumes of Similar Solids Objectives Determine the ratios of the areas and volumes of solids.

WARM UP IN NOTES Put the following aspects of solids into 1 of the 3 columns below (where does each go?) Side length, area of a base, radius, volume, diameter, circumference, height, lateral area, slant height, Perimeter, total area 1 Dimensional Measurement 2 Dimensional Measurement 3 Dimensional Measurement

If the scale factor of two similar figures is a:b, then… the ratio of their perimeters is a:b the ratio of their areas is a2:b2. ~ Scale Factor- 7: 3 Ratio of P – 7: 3 Ratio of A – 49 :9 14 6

What if these were similar Octoconal Prisims? Theorem If the scale factor of two similar solids is a:b, then… the ratio of their perimeters is a:b (1 dimension) the ratio of their areas is a2:b2 (2 dimensions) the ratio of their volumes is a3:b3 (3 dimensions) What if these were similar Octoconal Prisims? Scale Factor- 7: 3 Ratio of P – 7: 3 Ratio of A – 49 :9 Ratio of V – 343 :27 ~ 14 6

Similar Solids For two solids to be similar… The bases must be similar Same shape bases that are in proportion All corresponding measurements must be proportional. r h r h

Worksheet examples Got through example 1 on front and example 2 on back

White Boards The volumes of two spheres have a ratio of 27:64. Find the area of the larger sphere if the area of the smaller sphere is 18. 32

Similar Solids Scale Factor (1 dimension) Every measurement that is 1 dimension has a ratio that’s the same as the scale factor Side lengths, radii, diameters, circumference, heights, slant heights Perimeter If the ratio of the heights is 2:3 what is the ratio of the slant heights?

**WE MUST SQUARE THE SCALE FACTOR!!** Similar Solids Areas (2 dimensions) Measurements given in units2, cm2, in2 Areas of the corresponding bases Lateral areas Surface areas (total area) r h **WE MUST SQUARE THE SCALE FACTOR!!** r h 22:52  4:25 If the ratio of the radii are 4:10 what is the ratio of the base areas?

23:53  8:125 Similar Solids CUBE IT! Volume (3 dimensions) Measurements given in units3, cm3, in3 So what do you think we have to do with the scale factor? CUBE IT! r If the ratio of the radii are 4:10 what is the ratio of the volumes? r 23:53  8:125

Theorem If the scale factor of two similar solids is a:b, then… the ratio of their perimeters is a:b (1 dimension) the ratio of their areas is a2:b2 (2 dimensions) the ratio of their volumes is a3:b3 (3 dimensions) r r r r

White Boards Two regular pyramids have equilateral triangular bases with sides 4 and 6. Their heights are 6 and 9 respectively. Are the two pyramids similar? YES Scale Factor 2:3

White Boards Two similar cones have bases with area ratios of 4:9. Find the ratios of the following: Radii Heights Total areas Volumes 2:3 4:9 8:27

Perimeters 3:2 White Boards The volumes of two similar rectangular solids are 81 and 24. Find the ratio of their base perimeters. Perimeters 3:2 If the total area of the bigger is 36 find the total area of the smaller

White Boards The radii of two similar cylinders are 2 and 5. Find the ratios of their volumes and of their lateral areas. Volumes 8:125 Lateral areas 4:25