Unit 11 – Three Dimensional Geometry

Day Classwork Day Homework

Thursday

2/28 Unit 10 Test

Friday

3/1

Areas of Regular Polygons 1 HW 11.1

Monday

3/4

Volume of Prisms & Cylinders

2 HW 11.2

Tuesday

3/5

Volume of Pyramids and Cones

3 HW 11.3

Wednesday

3/6

Volume of Spheres

Unit 11 Quiz 1

4 HW 11.4

Thursday

3/7

Review with Word Problems

5 HW 11.5

Friday

3/8

Cavalieri’s Principle

Percent Error

Unit 11 Quiz 2

6 HW 11.6

Monday

3/11

The Scaling Principle for Volume

7 HW 11.7

Tuesday

3/12

Density and Dimensional Analysis 8 HW 11.8

Wednesday

3/13

Review

Unit 11 Quiz 3

9 Unit 11 Review Sheet

Thursday

3/14

Review

10 Unit 11 Review Sheet

Friday

3/15 No School – Staff Development Day

Monday

3/18

Review

11 Study!

Tuesday

3/19 Unit 12 Test 12

DAY 1 - AREAS OF REGULAR POLYGONS AND COMPOSITE FIGURES

Circumference and Area of a Circle Words Formula Model

The circumference of a circle is

equal to 2πr or πd.

The area of a circle equals πr2

1. Find the exact circumference and area of each circle.

a. b.

The center of a regular polygon and the radius of a regular polygon are also

the center and the radius of its circumscribed circle. A segment drawn perpendicular to a side of a regular polygon is called an apothem. A central angle of a regular polygon has its vertex at the center of the polygon and its

sides pass through consecutive vertices of the polygon.

1. Identify the center, a radius, an apothem, and a central angle of each polygon. Then find the measure of a central angle.

2. The cover of the hot tub shown is a regular pentagon. If the side length is 2.5 feet and the apothem is 1.7 feet, find the area of the lid to the nearest tenth.

3. Find the area of each regular polygon. Round to the nearest tenth.

a.

b.

c.

DAY 2 - DEFINITION AND PROPERTIES OF VOLUME

Volume is the measure of the space occupied by a solid. Standard measures of volume are cubic units such as cubic inches (in3) or cubic feet (ft3).

Area Properties Volume Properties

1. The area of a set in two dimensions is a number greater than or equal to zero that measures the size of the set and not the shape.

The volume of a set in three dimensions is a number greater than or equal to zero that measures the size of the set and not the shape.

2. The area of a rectangle is given by the formula length × width. The area of a triangle

is given by the formula 1

2× base × height. A

polygonal region is the union of finitely many non-overlapping triangular regions and has area the sum of the areas of the triangles.

A right rectangular or triangular prism has volume given by the formula area of base ×height. A right prism is the union of finitely many non-overlapping right rectangular or triangular prisms and has volume the sum of the volumes of the prisms.

3. Congruent regions have the same area. Congruent solids have the same volume.

4. The area of the union of two regions is the sum of the areas minus the area of the intersection:

Area(𝐴⋃𝐵) = Area(𝐴) + Area(𝐵) − Area(𝐴⋂𝐵)

The volume of the union of two solids is the sum of the volumes minus the volume of the intersection:

Vol(𝐴 ∪ 𝐵) = Vol(𝐴) + Vol(𝐵) − Vol(𝐴 ∩ 𝐵)

5. The area of the difference of two regions where one is contained in the other is the difference of the areas: If 𝐴 ⊆ 𝐵, then Area(𝐵 −𝐴) = Area(𝐵) − Area(𝐴).

The volume of the difference of two solids where one is contained in the other is the difference of the volumes:

If 𝐴 ⊆ 𝐵, then Vol(𝐵 − 𝐴) = Vol(𝐵) − Vol(𝐴).

6. The area 𝑎 of a region 𝐴 can be estimated by using polygonal regions 𝑆 and 𝑇 so that 𝑆 is contained in 𝐴 and 𝐴 is contained in 𝑇. Then Area(𝑆) ≤ 𝑎 ≤ Area(𝑇).

The volume 𝑣 of a solid 𝑊 can be estimated by using right prism solids 𝑆 and 𝑇 so that 𝑆 ⊆𝑊 ⊆ 𝑇. Then, Vol(𝑆) ≤ 𝑣 ≤ Vol(𝑇).

DAY 2 - VOLUME OF CYLINDERS & PRISMS

Figure and Description Sketch of Figure Sketch of Cross-Section

Right Prism

A general cylinder whose lateral edges are perpendicular to a polygonal base.

Oblique Prism

A general cylinder whose lateral edges are not perpendicular to a polygonal base.

Right Cylinder

A general cylinder whose lateral edges are perpendicular to a circular base.

Oblique Cylinder

A general cylinder whose lateral edges are not perpendicular to a circular base.

Volume of Prisms & Cylinders

Words Symbols Models

The volume V of prisms and cylinders is

V=Bh, where B is the area of the base and h is the height of the prism/cylinder.

Examples Find the volume of the following prisms.

1. 2.

3. 4.

5. Find the surface area of a rectangular solid with a length of 5cm, width of 8cm and a height

of 3cm. Draw a picture of a rectangular solid with your answer.

6.

7. In a sandcastle competition, contestants are allowed to use only water, shovels, and 10π cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be?

8. A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 cm tall, including 1 cm for the thickness of the base of the holder. This thickness of the rim of the holder is 1 cm. What is the volume of the rubberized material that makes up the holder?

9. Find the volume of the right pentagonal prism shown below.

10. A right prism has a rhombus for a base. The diagonals of the rhombus have lengths 10 and 14.

If the volume of the prism is 455 cubic units, what is the height of the prism?

11. Joe has a rectangular prism with a length of 10 centimeters, a width of 2 centimeters, and an

unknown height. He needs to build another rectangular prism with a length of 5 centimeters

and the same height as the original prism. The volume of the two prisms will be the same. Find

the width, in cm, of the new prism.

DAY 3 - VOLUME OF PYRAMIDS & CONES

Volume of a Pyramid Words Symbols Models

The volume of a pyramid is

V=(1/3)Bh, where B is the area of the base and h is the height of the

pyramid.

Examples Find the volume of the following pyramids.

1. 2.

3. At the top of the Washington Monument is a small square pyramid, called a pyramidion. This pyramid has a height of 55.5 feet with base edges of approximately 34.5 feet. What is the volume of the pyramidion? Round to the nearest tenth.

4. If a regular pyramid has a triangular base that is equilateral with sides measuring 4 feet, and its

volume is 364 3 ft , find the height of the pyramid.

Volume of a Cone

Words Symbols Models

The volume of a circular cone is V=(1/3)Bh, or V=(1/3)πr2h, where B is the area of the

base, h is the height of the cone, and r is the radius of the base.

Examples Find the volume of the following cones.

1. 2.

3. If a cone and cylinder have the same volume and the same height, and the cone has a radius of 9mm, find the radius of the cylinder.

4. If the slant height of a cone is 26, and the radius of the base is 10. Find the volume of the cone.

5. A cone fits inside a cylinder so that their bases are the same and their heights are the same, as shown in the diagram below. Calculate the volume that is inside the cylinder but outside of the cone. Give an exact answer.

6. A square pyramid has a volume of 245 in3. The height of the pyramid is 15 in. What is the area of the base of the pyramid? What is the length of one side of the base?

7. Use the diagram below to answer the questions that follow.

a. Determine the volume of the cone shown below. Give an exact answer.

b. Find the dimensions of a cone that is similar to the one given above. Explain how you found your answers.

c. Calculate the volume of the cone that you described in part (b) in two ways. (Hint: Use the volume formula and the scaling principle for volume.)

DAY 4 – VOLUME OF SPHERES

A sphere is the locus of all points in space that are a given distance from a given point called the center of the sphere.

Volume of a Sphere

Words Model

The volume V of a sphere is V= 4

3 πr3, where r is

the radius of a sphere.

A plane can intersect a sphere in a point or in a circle. If the circle contains the center of the sphere, the

intersection is called a great circle. The endpoints of a diameter of a great circle area called poles.

Intersection of Plane and Sphere

Examples Find the volume of each sphere or hemisphere. Round to the nearest tenth.

1. A hemisphere with a radius of 6 centimeters

2. A sphere with a great circle circumference of 18π centimeters.

3. If the volume of a sphere is 36π cubic feet, find the length of the diameter.

4. A plane P intersects a sphere with center O, forming a circle O’. If the plane is 8 inches from

center O and the area of circle O’ is 225π square inches, find (in terms of π) the number of cubic

inches in the volume of the sphere.

5. An ice cream cone is 11 cm deep and 5 cm across the opening of the cone. Two hemisphere-shaped scoops of ice cream, which also have diameters of 5 cm, are placed on top of the cone. If the ice cream were to melt into the cone, will it overflow?

6. Bouncy, rubber balls are composed of a hollow, rubber shell 0.4" thick and an outside diameter of 1.2". The price of the rubber needed to produce this toy is $0.035/in3.

a. What is the cost of producing 1 case, which holds 50 such balls? Round to the nearest cent.

b. If each ball is sold for $0.10, how much profit is earned on each ball sold?

7. The water tower in the picture below is modeled by the two-dimensional figure beside it. The

water tower is composed of a hemisphere, a cylinder, and a cone. Let C be the center of the

hemisphere and let D be the center of the base of the cone.

If feet, feet, and ,

determine and state, to the nearest cubic foot, the

volume of the water tower. The water tower was

constructed to hold a maximum of 400,000 pounds of

water. If water weighs 62.4 pounds per cubic foot, can

the water tower be filled to 85% of its volume and not

exceed the weight limit? Justify your answer.

DAY 6 - CAVALIERI’S PRINCIPLE

The bases of the following triangular prism 𝑇 and rectangular prism 𝑅 lie in the same plane. A plane that is parallel to the bases and also a distance 3 from the bottom base intersects both solids and

creates cross-sections 𝑇′ and 𝑅′.

a. Find Area(𝑇′).

b. Find Area(𝑅′).

c. Find Vol(𝑇).

d. Find Vol(𝑅).

e. If a height other than 3 were chosen for the cross-section, would the cross-sectional area of either solid change?

Are the two volumes different? Why?

1. Find the volume of the following oblique hexagonal prism.

2. Find the volume of an oblique cylinder that has a radius of 5 feet and a height of 3 feet. Round to the nearest tenth.

Cavalieri’s Principle Words Models

Given two solids that are included

between two parallel planes, if every plane parallel to the two planes intersects both solids in

cross-sections of equal area, then the volumes of the two solids are

equal.

Figure 3

FigurFig

Fig

3. Prisms A and B have the same length and width, but different heights. If the volume of Prism B is 150 cubic inches greater than the volume of Prism A, what is the length of each prism?

4. Morgan tells you that Cavalieri’s principle cannot apply to the cylinders shown below because their bases are different. Do you agree or disagree? Explain.

5. A triangular prism has an isosceles right triangular base with a hypotenuse of √32 and a prism height of 15. A square prism has a height of 15 and its volume is equal to that of the triangular prism. What are the dimensions of the square base?

Percent Error Words Formula

Relative error is the ratio of the absolute

error of the measurement/observed to the accepted measurement.

Percent of Error is found by multiplying the

relative error by 100%

1. The actual length of this field is 500 feet. A measuring instrument shows the length to be 508 feet. Find the percentage error in the measured length of the field.

2.

a. Approximate the area of a disk of radius 1 using an inscribed regular octagon. What is the percent error of the approximation?

(Remember that percent error is the absolute error as a percent of the exact measurement.)

b. Approximate the area of a circle of radius 1 using a circumscribed regular octagon. What is the percent error of the approximation?

3. John estimated that 500 cubic centimeters of salsa was missing from the jar shown below. Find the percent error of his estimation. Round to the nearest tenth of a percent.

4. A student’s measurement was found to have a 15.6% error. If his observed measurement was 25.7 mL, what are the two possible values for the actual measurement?

DAY 7 - THE SCALING PRINCIPLE FOR VOLUME

Each pair of solids shown below is similar. Write the ratio of side lengths 𝑎 ∶ 𝑏 comparing one pair of corresponding sides. Then, complete the third column by writing the ratio that compares volumes

of the similar figures. Simplify ratios when possible.

Similar Figures Ratio of Side

Lengths 𝒂 ∶ 𝒃

Ratio of Volumes

𝐕𝐨𝐥𝐮𝐦𝐞(𝑨) ∶ 𝐕𝐨𝐥𝐮𝐦𝐞(𝑩)

Figure A Figure B

Figure A

Figure B

Figure A Figure B

Figure A Figure B

Figure A

Figure B

Theorem

Words Examples Models

If two similar solids have a scale factor of a:b, then the surface areas

have a ratio of a2:b2, and the volumes have a ratio of a3:b3.

Scale factor: Ratio of Surface Areas: Ratio of Volumes:

Examples

1. Right circular cone 𝐴 has a volume of 675 units3 and a radius of 3. Right circular cone 𝐵 is similar to cone 𝐴 and has a volume of 1600 units3. Find the radius of cone 𝐵.

2. Two rectangular prisms are similar. The ratio of the areas of their bases is 25: 16. Find the ratio of the volumes of the similar solids.

3. The solids below are similar. Find the solid’s volume that is unknown. Round your answer to the

nearest tenth.

a. b.

1. Use the triangular prism shown to answer the questions that follow.

a. Calculate the volume of the triangular prism.

b. If one side of the triangular base is scaled by a factor of 2, the other side of the triangular base is scaled by a factor of 4, and the height of the prism is scaled by a factor of 3, what are the dimensions of the scaled triangular prism?

c. Calculate the volume of the scaled triangular prism.

d. Make a conjecture about the relationship between the volume of the original triangular prism and the scaled triangular prism.

2. In the following triangular pyramid, a plane passes through the pyramid so that it is parallel to the base and results in the cross-section △ 𝐴′𝐵′𝐶′. If the area of △ 𝐴𝐵𝐶 is 25 mm2, what is the area of △ 𝐴′𝐵′𝐶′?

3. In the following triangular pyramid, a plane passes through the pyramid so that it is parallel to the base and results in the cross-section △ 𝐴′𝐵′𝐶′. The altitude from 𝑉 is drawn; the intersection of the altitude with the base is 𝑋, and the intersection of the altitude with the cross-section is 𝑋′. If the distance from 𝑋 to 𝑉 is 18 mm, the distance from 𝑋′ to 𝑉 is 12 mm, and the area of △ 𝐴′𝐵′𝐶′ is 28 mm2, what is the area of △ 𝐴𝐵𝐶?

4. The diagram below shows a circular cone and a general pyramid (Cone). The bases of the cones are equal in area, and the solids have equal heights.

a. Sketch a slice in each cone that is parallel to

the base of the cone and 2

3 closer to the vertex

than the base plane.

b. If the area of the base of the circular cone is 616 units2, find the exact area of the slice drawn in the pyramid.

5. A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base) as shown below.

The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches, and the height of the glass is 5 inches. The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in order to find the height of the cone. Explain why. Determine and state, in inches, the height of the larger cone. Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.

DAY 8 – DENSITY & DIMENSIONAL ANALYSIS

1. A square metal plate has a density of 10.2 g/cm3 and weighs 2.193 kg.

a. Calculate the volume of the plate.

b. If the base of this plate has an area of 25 cm2, determine its thickness.

2. A metal cup full of water has a mass of 1,000 g. The cup itself has a mass of 214.6 g. If the cup has both a diameter and a height of 10 cm, what is the approximate density of water?

3. Calculate the density in g/cm3 of a cube with a volume of 256 cm3 that weighs 12 kilograms.

4. Calculate the volume of a right cylinder with a density of 0.4 kg/m3 that weighs 1500 grams.

We can use the formula density =mass

volume to find the density of a substance.

5. Gold has a density of 19.32 g/cm3. If a square pyramid has a base edge length of 5 cm, height of 6 cm, and a mass of 942 g, is the pyramid in fact solid gold? If it is not, what reasons could explain

why it is not? Recall that density can be calculated with the formula density =mass

volume.

6. New streetlights will be installed along a section of the highway. The posts for the streetlights will be 7.5 m tall and made of aluminum. The city can choose to buy the posts shaped like cylinders or the posts shaped like rectangular prisms. The cylindrical posts have a hollow core, with aluminum 2.5 cm thick, and an outer diameter of 53.4 cm. The rectangular-prism posts have a hollow core, with aluminum 2.5 cm thick, and a square base that measures 40 cm on each side. The density of aluminum is 2.7 g/cm3, and the cost of aluminum is $0.38 per kilogram. If all posts must be the same shape, which post design will cost the town less? How much money will be saved per streetlight post with the less expensive design?

7. About how many gallons of water does the watercooler bottle shown below contain?

(1 ft3 ≈ 7.5 gal)

8. Water flows at 2 feet per second through a pipe with a diameter of 8 inches. A cylindrical tank with a diameter of 15 feet and a height of 6 feet collects the water.

a. What is the volume, in cubic inches, of water flowing out of the pipe every second?

b. What is the height, in inches, of the water in the tank after 5 minutes?

c. How many minutes will it take to fill 75% of the tank?

9. You have 10 gallons of lemonade to sell. (1 gal ≈ 3785 cm3)

a. Each customer uses one paper cup. How many paper cups will you need?

b. The cups are sold in packages of 50. How many packages should you buy?

c. How many cups will be left over if you sell 80% of the lemonade?