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?

EXAMPLE 1 Identify and name polyhedra Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. a. b.

EXAMPLE 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.

EXAMPLE 1 Identify and name polyhedron 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.

GUIDED 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

GUIDED PRACTICE for Example 1 2. It has a curved square, so it is not a polyhedron. ANSWER

GUIDED 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

EXAMPLE 2 Use Euler’s Theorem in a real-world situation House Construction Find the number of edges on the frame of the house. 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.

Use Euler’s Theorem in a real-world situation EXAMPLE 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

Use Euler’s Theorem with Platonic solids EXAMPLE 3 Use Euler’s Theorem with Platonic solids Find the number of faces, vertices, and edges of the regular octahedron. Check your answer using Euler’s Theorem. SOLUTION By counting on the diagram, the octahedron has 8 faces, 6 vertices, and 12 edges. Use Euler’s Theorem to check. F + V = E + 2 8 + 6 = 12 + 2 14 = 14 Euler’s Theorem Substitute. This is a true statement. So, the solution checks.

EXAMPLE 4 Describe cross sections Describe the shape formed by the intersection of the plane and the cube. a. b. SOLUTION a. The cross section is a square. b. The cross section is a rectangle.

EXAMPLE 4 Describe cross sections c. SOLUTION c. The cross section is a trapezoid.

GUIDED 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. SOLUTION 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

GUIDED PRACTICE for Examples 2, 3, and 4 Describe the shape formed by the intersection of the plane and the solid. 5. ANSWER The cross section is a triangle

GUIDED PRACTICE for Examples 2, 3, and 4 6. ANSWER The cross section is a circle

GUIDED PRACTICE for Examples 2, 3, and 4 7. ANSWER The cross section is a hexagon

EXAMPLE 1 Find the number of unit cubes 3- D PUZZLE Find the volume of the puzzle piece in cubic units.

EXAMPLE 1 Find the number of unit cubes 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 is 14 + 4 + 2 = 20 cubic units.

EXAMPLE 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 1 2 = 30cm2 and h = 5 cm. V = Bh = 30(5) = 150cm3

EXAMPLE 2 Find volumes of prisms and cylinders SOLUTION b. Right cylinder b. The area of the base is π 92, or 81πft2. Use h = 6 ft to find the volume. V = Bh = 81π(6) = 486π ≈ 1526.81 ft3

The volume of the cube is 90 cubic inches. Find the value of x. EXAMPLE 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 Formula for volume of a cube 90 in3. = x3 Substitute for V. 4.48 in. ≈ x Find the cube root.

GUIDED 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

GUIDED 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

GUIDED PRACTICE for Example 1,2,and 3 3. The volume of a right cylinder is 684π cubic inches and the height is 18 inches. Find the radius. The radius of a right cylinder is in 38 ANSWER

Find the volume of an oblique cylinder EXAMPLE 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 Formula for volume of a cylinder = π(42)(7) Substitute known values. = 112π Simplify.

Find the volume of an oblique cylinder EXAMPLE 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

EXAMPLE 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.

Solve a real-world problem EXAMPLE 5 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

Solve a real-world problem EXAMPLE 5 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.

GUIDED 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

GUIDED PRACTICE for Example 4 and 5 5. Find the volume of the solid shown below. ANSWER 205.81 ft3

Daily Homework Quiz Determine whether the solid is a polyhedron. If it is, name it. 1. ANSWER no

Daily Homework Quiz 2. ANSWER yes; pyramid

Daily Homework Quiz 3. ANSWER yes; pentagonal prism

Daily Homework Quiz 4. Find the number of faces, vertices, and edges of each polyhedron in Exercises 1–3. ANSWER pyramid: 5 faces, 5 vertices, 8 edges; prism: 7 faces, 10 vertices, 15 edges 5. A plane intersects a cone, but does not intersect the base of the cone. Describe the possible cross sections. ANSWER a point, a circle, an ellipse

Daily 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

Daily 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?

Daily 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 about 7.2 in.

When is a solid a polyhedron? Essential Questions: When is a solid a polyhedron? How do you find the volume of a right prism of right cylinder? You will identify solids You will find volumes of prisms and cylinders • A solid is a polyhedron if it is bounded by polygons. A polyhedron is regular if all of its faces are congruent regular polygons. • For a polyhedron, F + V =E + 2. • The intersection of a plane and a solid is a cross section. If all the faces of a solid are polygons, then the solid is a polyhedron. • For a prism, V = Bh. • For a cylinder, V = Bh = π r 2h. • Cavalieri’s Principle says the volume formulas work for both right and oblique prisms and cylinders. The volume of a right prism or right cylinder is the product of the height and the area of the base.