12.1

Explore Solids Goal

Your Notes

p Identify solids.

VOCABULARY Polyhedron

Face Edge

Vertex

Base

Regular polyhedron

Convex polyhedron

Notice that the names of four of the Platonic solids end in “hedron.” Hedron is Greek for “side” or “face.” Sometimes a cube is called a regular hexahedron.

Platonic solids

Cross section

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325

12.1

Explore Solids Goal

Your Notes

p Identify solids.

VOCABULARY Polyhedron A polyhedron is a solid that is bounded by polygons that enclose a single region of space. Face The faces of a polyhedron are polygons. Edge An edge of a polyhedron is a line segment formed by the intersection of two faces. Vertex A vertex of a polyhedron is a point where three or more edges meet. Base A base is a polygon that is used to name the polyhedron. Regular polyhedron A regular polyhedron is a polyhedron whose faces are all congruent regular polygons. Convex polyhedron A convex polyhedron is a polyhedron such that any two points on its surface can be connected by a line segment that lies entirely inside or on the polyhedron.

Notice that the names of four of the Platonic solids end in “hedron.” Hedron is Greek for “side” or “face.” Sometimes a cube is called a regular hexahedron.

Platonic solids A Platonic solid is one of five regular polyhedra: a regular tetrahedron, a cube, a regular octahedron, a regular dodecahedron, and a regular icosahedron. Cross section A cross section is the intersection of a plane and a solid.

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Your Notes

TYPES OF SOLIDS Polyhedra

Not Polyhedra

Prism

Cylinder

Pyramid

Example 1

Cone

Sphere

Describe polyhedra

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

b.

c.

Solution a. This is a polyhedron. It has faces so it is a . It has vertices and edges. b. This is a polyhedron. The two bases are congruent hexagons, so it is a . It has faces, vertices, and edges. c. This is not a polyhedron. The solid has a curved surface.

THEOREM 12.1: EULER’S THEOREM The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F1V5E1 326

Lesson 12.1 • Geometry Notetaking Guide

.

F 5 6, V 5 8, E 5 12 6 1 8 5 12 1 2

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Your Notes

TYPES OF SOLIDS Polyhedra

Not Polyhedra

Prism

Cylinder

Pyramid

Example 1

Cone

Sphere

Describe polyhedra

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

b.

c.

Solution a. This is a polyhedron. It has 4 faces so it is a tetrahedron . It has 4 vertices and 6 edges. b. This is a polyhedron. The two bases are congruent hexagons, so it is a hexagonal prism . It has 8 faces, 12 vertices, and 18 edges. c. This is not a polyhedron. The solid has a curved surface.

THEOREM 12.1: EULER’S THEOREM The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F1V5E1 2 . 326

Lesson 12.1 • Geometry Notetaking Guide

F 5 6, V 5 8, E 5 12 6 1 8 5 12 1 2

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Your Notes

Example 2

Use Euler’s Theorem

Find the number of faces, vertices, and edges of the polyhedron shown. Check your answers using Euler’s Theorem. The polyhedron has and edges.

faces,

vertices,

Use Euler’s Theorem to check. F1V5E12 1

5

12

5

Example 3

Use a 30°–60°– 90° triangle to find the height of the triangle in part a. Use a 45°–45°–90° triangle to find the length of a diagonal in part b. Apply the Pythagorean Theorem to find the other lengths you need for parts b and c.

Euler’s Theorem Substitute. Check.

Describe cross sections

Identify and sketch the cross section of the triangular prism. Then find the perimeter and area of the cross section. Every edge of the prism is 4 cm long. a.

b.

P5 A5 c.

d.

P5 A5

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P5 A5

P5 A5

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327

Your Notes

Example 2

Use Euler’s Theorem

Find the number of faces, vertices, and edges of the polyhedron shown. Check your answers using Euler’s Theorem. The polyhedron has 8 faces, 12 vertices, and 18 edges. Use Euler’s Theorem to check. F1V5E12

Euler’s Theorem

8 1 12 5 18 1 2

Substitute.

20 5 20

Example 3

Use a 30°–60°– 90° triangle to find the height of the triangle in part a. Use a 45°–45°–90° triangle to find the length of a diagonal in part b. Apply the Pythagorean Theorem to find the other lengths you need for parts b and c.

Check.

Describe cross sections

Identify and sketch the cross section of the triangular prism. Then find the perimeter and area of the cross section. Every edge of the prism is 4 cm long. Isosceles a. b. triangle Equilateral triangle 4 cm 2

3 cm 4 cm

4 cm

P 5 12 }cm A 5 4Ï 3 cm2 c.

4

4 2 cm

Isosceles trapezoid

d.

2

2 cm

7 cm 4 cm

}

P 5 4 1} 8Ï 2 cm A 5 4Ï 7 cm2 Rectangle

2 cm 2 5 cm

19 cm

4 cm

2 5 cm

}

P 5 6 1}4Ï 5 cm A 5 3Ï 19 cm2

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4 cm

2 cm

P 5 12 cm A 5 8 cm2

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Your Notes

Checkpoint Complete the following exercises.

In Exercises 1–3, tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 1.

2.

3.

4. Is it possible for a polyhedron to have 16 faces, 34 vertices, and 50 edges? Explain.

5. Identify and sketch the cross section of the cylinder. Then find the circumference and area of the cross section. The diameter of the cylinder is 12 cm and its height is 20 cm.

Homework

328 Lesson 12.1 • Geometry Notetaking Guide

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Your Notes

Checkpoint Complete the following exercises.

In Exercises 1–3, tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 1.

This is a polyhedron. Its bases are congruent pentagons, so it is a pentagonal prism. It has 7 faces, 10 vertices, and 15 edges.

2.

This is not a polyhedron. The solid has a curved surface.

3.

This is a polyhedron. Its base is a rectangle, so it is a rectangular pyramid. It has 5 faces, 5 vertices, and 8 edges.

4. Is it possible for a polyhedron to have 16 faces, 34 vertices, and 50 edges? Explain. No; Using Euler’s Theorem F 1 V 5 E 1 2, 16 1 34 Þ 50 1 2. 5. Identify and sketch the cross section of the cylinder. Then find the circumference and area of the cross section. The diameter of the cylinder is 12 cm and its height is 20 cm. Circle 6 cm

Homework C 5 12π cm A 5 36π cm2

328 Lesson 12.1 • Geometry Notetaking Guide

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