Explore Solids

Key Concepts

• Identify and name polyhedral.
• Use Euler’s Theorem in a real-world situation.
• Use Euler’s Theorem with platonic solids.
• Describe cross sections.

Introduction 

Polyhedron

A polyhedron is a solid that is bounded by polygons called faces that enclose a single region of space. It is a three-dimensional solid made up of plane faces. Poly=many Hedron=faces. 

An edge of a polyhedron is a line segment formed by the intersection of two faces of Explore Solids.   

A vertex of a polyhedron is a point where three or more edges meet. The plural of polyhedron is polyhedra or polyhedrons. 

Prism

Polyhedron with two parallel, congruent bases. Named after its base. 

Pyramid

Polyhedron with one base and lateral faces. Named after its base. 

Identify and name polyhedra

Example 1: 

Decide whether the solid is a polyhedron. If so, count the number of faces, vertices, and edges of the polyhedron.   

Solution: 

1. This is a polyhedron. It has 5 faces, 6 vertices, and 9 edges. 
2. This is not a polyhedron. Some of its faces are not polygons. 
3. This is a polyhedron. It has 7 faces, 7 vertices, and 12 edges. 

Use Euler’s Theorem in a real-world situation 

The sum of the number of faces and vertices is two more than the number of edges in the solids above. This result was proved by the Swiss mathematician Leonhard Euler. 

As per Euler’s Theorem, 

• The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula:  

  F + V = E + 2 

Example 2: 

The box shown at the right is a hexagonal prism. It has 8 faces. Two faces are hexagons, and 6 faces are squares. Count the edges and vertices. Use Euler’s Theorem to check your answer. 

Solution: 

On their own, 2 hexagons and 6 squares have 2(6) + 6(4), or 36 edges. In the solid, each side is shared by exactly two polygons. So the number of edges is one half of 36, or 18. Use Euler’s Theorem to find the number of vertices. 

F + V = E + 2 Write Euler’s Theorem. 

8 + V = 18 + 2 Substitute values. 

8 + V = 20 Simplify. 

V = 12 Solve for V. 

The box has 12 vertices. 

Use Euler’s Theorem with Platonic solids 

Types of Solids

Of the first solids below. The prism and pyramid are polyhedra.  

The cone, cylinder, and sphere are not polyhedra. 

Regular/Convex/Concave

A polyhedron is regular if all its faces are congruent regular polygons. A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. 

If this segment goes outside the polyhedron, then the polyhedron is said to be NON-CONVEX OR CONCAVE. 

Platonic solids

There are five regular polyhedra called Platonic solids, after the Greek mathematician and philosopher Plato. 

The Platonic solids are a regular tetrahedron (4 faces), a cube (6 faces), a regular octahedron  
(8 faces), a regular dodecahedron (12 faces), and a regular icosahedron (20 faces). 

Example 3: 

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

Solution: 

By counting on the diagram, the tetrahedron has 4 faces, 4 vertices, and 6 edges. Use Euler’s Theorem to check. 

F + V = E + 2 Write Euler’s Theorem. 

4 + 4 = 6 + 2 Substitute values. 

8 = 8 This is true statement. So, the solution checks. 

Describe cross sections 

Cross section

The intersection of the plane and the solid is called a cross section. 

For instance, the diagram shows that the intersection of a plane and a sphere is a circle. 

Example 4: 

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

Solution: 

1. The cross section is a square. 
2. The cross section is a pentagon. 
3. The cross section is a triangle. 

Exercise

• ____________________ is a three-dimensional solid made up of plane faces.
• Decide whether the solid is a polyhedron. If so, count the number of faces, vertices, and edges of the polyhedron.

• Determine if the following solids are convex or concave.

• Describe the shape formed by the intersection of the plane and the closed cylinder.
• Use Euler’s Theorem to find the value of n.

Faces: n

Vertices: 8

Edges: 12

• Use Euler’s Theorem to find the value of n.

Faces: 8

Vertices: n

Edges: 18

• Name the five Platonic solids and give the number of faces for each.
• Find the number of faces, vertices, and edges of the polyhedron. Check your answer using Euler’s Theorem.                      

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

• The speaker shown at the right has 7 faces. Two faces are pentagons, and 5 faces are rectangles.

Find the number of vertices.

Use Euler’s Theorem to determine how many edges the speaker has.

Concept Map

What have we learned

• Using properties of polyhedra.
• Using Euler’s theorem.
• Describe cross sections by looking at the intersection of two shapes.