GEOMETRY
UNIT 3
WORKBOOK
CHAPTER 12
Surface Area & Volume
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Geometry
Section 11.4 Notes: Areas of Regular Polygons and Composite Figures
Label the Parts:
Steps for Finding the Area of a Regular Polygon:
1. Split the shape up into ________________.
2. Find the measure of the __________________
____________________.
Center
Radius
Apothem
Central Angle
3. Cut the triangle in _______________.
4. Solve for the _____________ and/or
_________________ of the triangle.
5. Determine the ____________ of one triangle.
6. Multiply the area of one triangle by the
_____________________ number of triangles.
Example 1: In the figure, pentagon PQRST is inscribed in circle X. Identify the center, a radius, an apothem, and a central
angle of the polygon. Then find the measure of a central angle.
Example 2: a) Find the area of the regular hexagon. Round to the nearest tenth.
b) Find the area of the regular pentagon. Round to the nearest tenth.
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Composite Figures
Example 4: a) The dimensions of an irregularly shaped pool are shown. What is the area of the surface of the pool?
b) Find the area of the figure in square feet. Round to the nearest tenth if necessary.
Example 5: a) Find the area of the shaded figure.
b) Cara wants to wallpaper one wall of her family room. She has a fireplace in the center of the wall. Find the area of the
wall around the fireplace.
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Geometry
11.4 Day 1 Homework
Name:_________________________________
Answer the following questions in order to find the area of each regular polygon.
1)
2)
1) What type of shape is this?
1) What type of shape is this?
2) What is the measure of a central angle?
2) What is the measure of the top angle of the triangle after
cutting it in half?
3) What is the measure of the top angle of the triangle after
cutting it in half?
3) What is the apothem/height of the triangle?
4) What is the apothem/height of the triangle?
4) What is the whole side length of the polygon?
5) What is half the side length of the polygon?
5) What is the area of one triangle?
6) What is the whole side length of the polygon?
6) Find the Area of the entire shape. Give an exact answer and
an answer rounded to the nearest hundredth.
7) What is the area of one triangle?
8) Find the Area of the entire shape to the nearest hundredth.
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Find the area of the regular polygon. Round your answer to the nearest hundredth.
3)
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Geometry
11.4 Day 2 Homework
Name:___________________________
Find the area of each regular polygon. Round your answers to the nearest hundredth.
1)
2)
Find the area of each figure. Round to the nearest hundredth.
3)
4)
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Geometry
Section 12.1 Notes: Cross Sections
Example 1: Scientists are able to use computers to study cross sections of ancient artifacts and structures. Determine the
shape of each cross section of the pyramid below:
Example 2: A customer ordered a two-layer sheet cake. Determine the shape of each cross section of the cake below.
Example 3: Ramona has a cake pan shaped like half a sphere, as shown. Describe the shape of the cross sections of cakes
baked in this pan if they are cut horizontally and vertically.
Example 4: Name the different cross sections of a cylinder. State how the plane would have to intersect the cylinder to get
that particular cross section.
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Practice
1.
A square pyramid is cut along the shaded plane
shown below. Draw a picture of the
cross-section of this solid.
5. A rectangular prism is cut along the shaded plane
shown below. Draw a picture
of the cross-section of this solid.
2. A cube with a cylinder cut from its center is
cut along the plane shown. Draw a picture of
the cross-section of this solid?
6. A cross-section is cut from the cylinder below.
What is the shape of the cross-section?
3. A cross section is cut from the circular cone
below. What is the shape of the cross-section?
7. What is the cross section of a sphere?
8. A square pyramid is cut along the shaded plane
show below. What is the shape of the cross-section of
this solid?
4. A cube with a cylinder cut from its center is cut
along the plane shown below. Draw a picture of
the cross-section of this solid.
9. What is the cross section of a cone cut
vertically?
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Geometry
Section 12.1 Worksheet
Name: _____________________________________
Circle the correct answer for each question.
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3
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5 Choose an object from your house. Describe the different cross sections the can be formed with this object.
Object: ______________________
Cross Sections:
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Geometry
Section 12.2 Notes: Surface Areas of Prisms and Cylinders
Prisms
Example 1: Find the lateral area and total surface area of the prisms below.
a)
b)
Example 2: Find the lateral area and total surface area of the prisms below.
a)
b)
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Example 3: Find the lateral area and total surface area of the rectangular prism.
Cylinders
Example 4: Find the lateral area and the total surface area of the cylinder.
Example 5: A can of soup is covered with the label shown. What is the radius of the soup can?
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Geometry
Section 12.2 Worksheet
Name: _____________________________________
For numbers 1 – 6, find the lateral area and total surface area of each prism. Round to the nearest tenth if necessary. Leave answers
exact when you can.
1.
2.
Lateral Area: __________________
Lateral Area: __________________
Surface Area: __________________
Surface Area: __________________
3.
4.
Lateral Area: __________________
Lateral Area: __________________
Surface Area: __________________
Surface Area: __________________
5.
6.
Lateral Area: __________________
Lateral Area: __________________
Surface Area: __________________
Surface Area: __________________
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For numbers 7 – 10, find the lateral area and surface area of each cylinder. Leave your answer in terms of pi and round to the nearest
tenth.
7.
8.
Lateral Area: __________________
Lateral Area: __________________
Surface Area: __________________
Surface Area: __________________
9.
10.
Lateral Area: __________________
Lateral Area: __________________
Surface Area: __________________
Surface Area: __________________
11. CAKES A cake is a rectangular prism with height 4 inches and base 12 inches by 15 inches. Wallace wants to apply frosting to
the sides and the top of the cake. What is the surface area of the part of the cake that will have frosting?
12. EXHAUST PIPES An exhaust pipe is shaped like a cylinder with a height of 50 inches and a radius of 2 inches. What is the
lateral surface area of the exhaust pipe? Round your answer to the nearest hundredth.
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Geometry
Section 12.3 Notes: Surface Areas of Pyramids and Cones
Pyramid and its parts:
Example 1: Find the lateral area of the square pyramid.
Example 2: Find the surface are of the square pyramid.
Example 3: Find the surface area of the regular pyramid.
Think about it…How would you find the total surface area of the composite figure?
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Cone and its parts:
Example 4: Find the surface area of the cone.
Example 5: A sugar cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the sugar cone.
Example 6: Find the total surface area of the solid.
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Geometry
Section 12.3 Worksheet
Name: _____________________________________
For numbers 1 – 4, find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary. Leave
answers exact if you can.
1.
2.
Lateral Area: ____________________
Lateral Area: ____________________
Surface Area: ____________________
Surface Area: ____________________
3.
4.
Lateral Area: ____________________
Lateral Area: ____________________
Surface Area: ____________________
Surface Area: ____________________
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For numbers 5 & 6, find the lateral area and surface area of each cone. Round to the nearest tenth if necessary.
5.
6.
Lateral Area: ____________________
Lateral Area: ____________________
Surface Area: ____________________
Surface Area: ____________________
7. Find the surface area of a cone if the height is 14 centimeters and the slant height is 16.4 centimeters.
8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.
9. Wendy bought a conical hat on a recent trip to central Vietnam. The basic frame of the hat is 16 hoops of bamboo that gradually
diminish in size. The hat is covered in palm leaves. If the hat has a diameter of 50 centimeters and a slant height of 32 centimeters,
what is the lateral area of the conical hat?
10. Find the total surface area of the composite solid.
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Geometry
Section 12.4 Notes: Volumes of Prisms and Cylinders
Example 1: Find the volume of the prism. Make sure to label the correct units.
a)
b)
Example 2: Find the volume of the cylinder. Make sure to label the correct units.
a)
b)
Example 3: Find the volume of the oblique cylinder. Label your units.
a)
b)
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Example 4: Prisms A and B have the same width and height, but different lengths. If the volume of Prism B is 128 cubic
inches greater than the volume of Prism A, what is the length of each prism?
Prism A
Prism B
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Geometry
Section 12.4 Worksheet
Name: _____________________________________
For numbers 1 -6, find the volume of each prism or cylinder. Round to the nearest tenth if necessary. Leave exact answers when
possible.
1.
2.
Volume: _________
Volume: _________
3.
4.
Volume: _________
Volume: _________
5.
6.
Volume: _________
Volume: _________
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7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25.5 inches and a radius
of 21 inches.
Note: 1 ft3 = 1728 in3
a) What is the volume of the aquarium in cubic feet?
b) If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium hold?
c) If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in the aquarium to the "nearest five pounds?
8. TRASH CANS The Meyer family uses a kitchen trash can shaped like a cylinder. It has a height of 18 inches and a base diameter
of 12 inches. What is the volume of the trash can? Round your answer to the nearest tenth of a cubic inch.
9. BENCH Inside a lobby, there is a piece of furniture for sitting. The furniture is shaped like a simple block with a square base 6 feet
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on each side and a height of feet. What is the volume of the seat?
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10. FRAMES Margaret makes a square frame out of four pieces of wood. Each piece of wood is a rectangular prism with a length of
40 centimeters, a height of 4 centimeters, and a depth of 6 centimeters. What is the total volume of the wood used in the frame?
11. PENCIL GRIPS A pencil grip is shaped like a triangular prism with a cylinder removed from the middle. The base of the prism is
a right isosceles triangle with leg lengths of 2 centimeters. The diameter of the base of the removed cylinder is 1 centimeter. The
heights of the prism and the cylinder are the same, and equal to 4 centimeters. What is the exact volume of the pencil grip?
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Geometry
Section 12.5 Notes: Volumes of Pyramids and Cones
Example 1: Find the volume of the square pyramid. Label the units.
a)
b)
Example 2: Find the volume of the cone. Label the units.
a)
b)
Example 3: A gazebo has a pentagonal base with an area of 80 square meters. The total height to the peak is 8 meters.
The height of the pyramidal roof is 2 meters. Find the gazebo’s total volume.
2m
8m
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Example 4: The volume of a right cone is 1350𝜋 m3 and the radius is 18 meters. Find the height of the cone.
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Geometry
Section 12.5 Worksheet
Name: _____________________________________
For numbers 1 – 6, find the volume of each pyramid or cone. Round to the nearest tenth if necessary. Leave answer exact if possible.
1.
2.
Volume: _________
Volume: _________
3.
4.
Volume: _________
Volume: _________
5.
6.
Volume: _________
Volume: _________
7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in diameter and the height of the shed
is 3.8 meters. What is the volume of the shed?
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8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the 27th century B.C. In its original state, it
stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid.
9. SCULPTING A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the
base of the cone and cylinder is 2 feet. What is the volume of the stone that the sculptor must remove? Round your answer to the
nearest hundredth.
10. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the
top square is 12 feet and the side length of the bottom square is 16 feet. The height of the stage is 3 feet.
a) What is the volume of the entire square pyramid that the stage is part of?
b) What is the volume of the top of the pyramid that is removed to get the stage?
c) What is the volume of the stage?
11. Find the volume of the solid (a rectangular prism with a right cone cut out of it). Round to the nearest tenth.
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Geometry
Section 12.6 Notes: Surface Areas and Volumes of Spheres
Example 1: Find the surface area and volume of the sphere.
a)
b)
Example 2: Find the surface area and volume of the sphere given the circumference or area of the great circle.
a)
b)
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Example 3: a) Find the lateral area, total surface area, and volume of the hemisphere. Label the units.
a)
b)
Example 4: Find the surface area (to the nearest tenth) and volume of the cone (in terms of pi).
Example 5: a)The stone spheres of Costa Rica were made by forming granodiorite boulders into spheres. One of the stone
spheres has a volume of about 36,000π cubic inches. What is the diameter of the stone sphere?
b) The jungle gym outside of Jada’s school is a perfect hemisphere. It has a volume of 4,000π cubic feet. What is the
diameter of the jungle gym?
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Geometry
Section 12.6 Worksheet
Name: _____________________________________
For numbers 1 – 4, find the surface area of each sphere or hemisphere. Round to the nearest tenth. Leave answers exact when
possible.
1.
2.
Surface Area: ____________________
Surface Area: ____________________
3. hemisphere: radius of great circle = 8.4 in.
4. sphere: area of great circle ≈ 29.8 m2
Surface Area: ____________________
Surface Area: ____________________
For numbers 5 – 9, find the volume of each sphere or hemisphere. Round to the nearest tenth.
5.
6.
Volume: ____________________
Volume: ____________________
7. hemisphere: diameter = 18 mm
7. Volume: ____________
8. sphere: circumference ≈ 36 yds
8. Volume: ____________
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9. Find the total surface area and volume of the composite solid. Leave your answer in terms of pi.
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10. BILLIARDS A billiard ball set consists of 16 spheres, each 2 inches in diameter. What is the total volume of a complete set of
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billiard balls? Round your answer to the nearest thousandth of a cubic inch.
11. MOONS OF SATURN The planet Saturn has several moons. These can be modeled accurately by spheres. Saturn’s largest moon
Titan has a radius of about 2575 kilometers. What is the approximate surface area of Titan? Round your answer to the nearest tenth.
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