11.5 Explore Solids&

11.6 Volume of Prisms and Cylinders• You will identify solids• You will find volumes

of prisms and cylinders

Essential Questions:• When is a solid a

polyhedron?• How do you find the

volume of a right prism of right cylinder?

Warm-Up ExercisesEXAMPLE 1

Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges.

a. b.

Identify and name polyhedra

Warm-Up ExercisesEXAMPLE 1 Identify and name polyhedron

c.

SOLUTION

The solid is formed by polygons, so it is a polyhedron. The two bases are congruent rectangles, so it is a rectangular prism. It has 6 faces, 8 vertices, and 12 edges.

a.

Warm-Up ExercisesEXAMPLE 1

The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. It has 7 faces, consisting of 1 base, 3 visible triangular faces, and 3 non-visible triangular faces. The polyhedron has 7 faces, 7 vertices, and 12 edges.

b.

The cone has a curved surface, so it is not a polyhedron.

c.

Identify and name polyhedron

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges.

1.

The solid is formed by polygons so it is a polyhedron. The base is a square it has 5 faces, 5 vertices and 8 edges.

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 1

2.

It has a curved square, so it is not a polyhedron.

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 1

3.

The solid is formed by polygons, so it is a polyhedron. The base is a triangle, so it is a triangular prism. It has 5 face, 6 vertices, and 9 edges.

ANSWER

Warm-Up ExercisesEXAMPLE 2 Use Euler’s Theorem in a real-world situation

SOLUTION

The frame has one face as its foundation, four that make up its walls, and two that make up its roof, for a total of 7 faces.

Find the number of edges on the frame of the house.

House Construction

Warm-Up ExercisesEXAMPLE 2 Use Euler’s Theorem in a real-world situation

To find the number of vertices, notice that there are 5 vertices around each pentagonal wall, and there are no other vertices. So, the frame of the house has 10 vertices.

Use Euler’s Theorem to find the number of edges.

F + V = E + 2

7 + 10 = E + 2

15 = E

Euler’s Theorem

Substitute known values.

Solve for E.

The frame of the house has 15 edges.ANSWER

Warm-Up ExercisesEXAMPLE 3 Use Euler’s Theorem with Platonic solids

SOLUTION

By counting on the diagram, the octahedron has 8 faces, 6 vertices, and 12 edges. Use Euler’s Theorem to check.

Find the number of faces, vertices, and edges of the regular octahedron. Check your answer using Euler’s Theorem.

F + V = E + 2

8 + 6 = 12 + 2

14 = 14

Euler’s Theorem

Substitute.

This is a true statement. So, the solution checks.

Warm-Up ExercisesEXAMPLE 4

Describe the shape formed by the intersection of theplane and the cube.

a. b.

Describe cross sections

b. The cross section is a rectangle.

a. The cross section is a square.

SOLUTION

Warm-Up ExercisesEXAMPLE 4 Describe cross sections

SOLUTION

c. The cross section is a trapezoid.

c.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

4. Find the number of faces, vertices, and edges of the regular dodecahedron on page 796. Check

your answer using Euler’s Theorem.

Counting on the diagram, the dodecahedron has 12 faces, 20 vertices, and 30 edges. Use Euler’s theorem to

F + V = E + 2

12 + 20 = 30 + 2

32 = 32

Check Euler’s theorem

Substitute

This is a true statement so, the solution check

SOLUTION

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

Describe the shape formed by the intersection of theplane and the solid.

5.

ANSWER

The cross section is a triangle

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

6.

ANSWER

The cross section is a circle

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

7.

ANSWER

The cross section is a hexagon

Warm-Up ExercisesEXAMPLE 1 Find the number of unit cubes

3- D PUZZLE Find the volume of the puzzle piece in cubic units.

Warm-Up ExercisesEXAMPLE 1

SOLUTION

To find the volume, find the number of unit cubes it contains. Separate the piece into three rectangular boxes as follows:

The base is 7 units by 2 units. So, it contains 7 . 2, or 14 unit cubes. The upper left box is 2 units by 2 units. So, it contains 2 . 2, or 4 unit cubes. The upper right box is 1 unit by 2 units. So, it contains 1 . 2, or 2 unit cubes.

By the Volume Addition Postulate, the total volume of the puzzle piece is14 + 4 + 2 = 20 cubic units.

Find the number of unit cubes

Warm-Up ExercisesEXAMPLE 2 Find volumes of prisms and cylinders

Find the volume of the solid.

a. Right trapezoidal prism

SOLUTION

(3)(6 + 14)a. The area of a base is 12

= 30cm2 and h = 5 cm.

V = Bh = 30(5) = 150cm3

Warm-Up ExercisesEXAMPLE 2 Find volumes of prisms and cylinders

b. Right cylinder

SOLUTION

b. The area of the base is

V = Bh = 81π(6) = 486π ≈ 1526.81 ft3

π 92, or 81πft2. Use h = 6 ft to find the volume.

Warm-Up ExercisesEXAMPLE 3 Use volume of a prism

ALGEBRA

The volume of the cube is 90 cubic inches. Find the value of x.

SOLUTION

A side length of the cube is x inches.

V = x3

90 in3. = x3

4.48 in. ≈ x

Formula for volume of a cube

Substitute for V.

Find the cube root.

Warm-Up ExercisesGUIDED PRACTICE for Example 1,2,and 3

1. Find the volume of the puzzle piece shown in cubic units.

The volume of the puzzle cube is 7 units3

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 1,2,and 3

2. Find the volume of a square prism that has a base edge length of 5 feet and a height of 12 feet.

The volume of a square prism is 300 ft3

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 1,2,and 3

3. The volume of a right cylinder is 684π cubicinches and the height is 18 inches. Find the radius.

The radius of a right cylinder is in 38

ANSWER

Warm-Up ExercisesEXAMPLE 4 Find the volume of an oblique cylinder

Find the volume of the oblique cylinder.

SOLUTION

Cavalieri’s Principle allows you to use Theorem 12.7 to find the volume of the oblique cylinder.

V = π r2h

= π(42)(7) Substitute known values.

= 112π Simplify.

Formula for volume of a cylinder

Warm-Up ExercisesEXAMPLE 4 Find the volume of an oblique cylinder

≈ 351.86 Use a calculator.

The volume of the oblique cylinder is about 351.86 cm3.

ANSWER

Warm-Up ExercisesEXAMPLE 5 Solve a real-world problem

PLANTER The planter is made up of 13 beams. In centimeters, suppose the dimensions of each beam are 30 by 30 by 90. Find its volume.

Warm-Up ExercisesEXAMPLE 5 Solve a real-world problem

SOLUTION

The area of the base B can be found by subtracting the area of the small rectangles from the area of the large rectangle.

B = Area of large rectangle – 4 Area of small rectangle

= 90 510 – 4(30 90)

= 35,100 cm2

Warm-Up ExercisesEXAMPLE 5 Solve a real-world problem

Use the formula for the volume of a prism.

V = Bh Formula for volume of a prism

= 35,100(30) Substitute.

= 1,053,000 cm3 Simplify.

The volume of the planter is 1,053,000 cm3, or 1.053 m3.

Warm-Up ExercisesGUIDED PRACTICE for Example 4 and 5

4. Find the volume of the oblique prism shown below.

The volume of the oblique prism is 180m3

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 4 and 5

5. Find the volume of the solid shown below.

ANSWER

205.81 ft3

Warm-Up ExercisesDaily Homework Quiz

Determine whether the solid is a polyhedron. If it is, name it.

1.

ANSWER

no

Warm-Up ExercisesDaily Homework Quiz

2.

ANSWER

yes; pyramid

Warm-Up ExercisesDaily Homework Quiz

3.

ANSWER

yes; pentagonal prism

Warm-Up ExercisesDaily Homework Quiz

4. Find the number of faces, vertices, and edges

of each polyhedron in Exercises 1–3.

5. A plane intersects a cone, but does not intersect the base of the cone. Describe the possible cross sections.

ANSWER

pyramid: 5 faces, 5 vertices, 8 edges;prism: 7 faces, 10 vertices, 15 edges

ANSWER

a point, a circle, an ellipse

Warm-Up ExercisesDaily Homework Quiz

1. Find the volume of the solid at the right.

ANSWER 5437.5 in.3

2. Find the volume of a right triangular prism with height 32 in., base height 12 in., and base length 18 in.

ANSWER 3456 in.3

Warm-Up ExercisesDaily Homework Quiz

3. Find the volume of a right cylinder with height 30 ft and diameter 14 ft.

ANSWER

4618.14 in.3

4. A cylindrical beaker with diameter 10 in. and height 12 in. is filled with water that is then poured into a rectangular pan that is 14 in. by 9 in. by 3 in. What is the volume of each solid? Would the water overflow the pan? If so, what is the height of the water in the beaker after the pan is filled?

Warm-Up ExercisesDaily Homework Quiz

ANSWER

cylinder: 942.48 in.3; pan: 378 in.3; Yes, it would overflow the pan. After the pan is filled , the height of the water in the beaker will be about7.2 in.

• You will identify solids

• You will find volumes of prisms and cylinders

Essential Questions:

• When is a solid a polyhedron?

• How do you find the volume of a right prism of right cylinder?• For a prism, V = Bh.

• For a cylinder, V = Bh = π r 2h.• Cavalieri’s Principle says thevolume formulas work for bothright and oblique prisms andcylinders.

The volume of a right prism orright cylinder is the product of the height and the area of the base.

• A solid is a polyhedron if it isbounded by polygons. A polyhedronis regular if all of its facesare congruent regular polygons.• For a polyhedron, F + V =E + 2.• The intersection of a plane and asolid is a cross section.

If all the faces of a solid are polygons, then the solid is a polyhedron.