Three-Dimensional Figures - Three-Dimensional Figures - High School Geometry Unlocked (2016)

High School Geometry Unlocked (2016)

Chapter 5. Three-Dimensional Figures

Lesson 5.2. Three-Dimensional Figures

POLYHEDRA

A polyhedron (plural polyhedra) is a three-dimensional shape for which each face is a flat surface. In other words, each face is a polygon. The faces intersect each other as straight line segments, and these edges intersect each other at single-point vertices.

A regular polyhedron is one whose faces are all congruent, regular polygons. Also, in a regular polyhedron, the same number of faces meet at each vertex. There are only five types of regular, convex polyhedra, and they have special names (see the figures and descriptions on the next page).

Tetrahedron

Hexahedron (Cube)

Octahedron

Dodecahedron

Icosahedron

Tetrahedron—4 faces, which are equilateral triangles. Three triangles meet at each vertex.

Hexahedron (Cube)—6 faces, which are squares. Three squares meet at each vertex.

Octahedron—8 faces, which are equilateral triangles. Four triangles meet at each vertex.

Dodecahedron—12 faces, which are regular pentagons. Three pentagons meet at each vertex.

Icosahedron—20 faces, which are equilateral triangles. Five triangles meet at each vertex.

Why only five? Consider what happens when you connect two or more polygons at a single vertex. If you connect only two polygons, then you wouldn’t have a solid. Therefore, to make a polyhedron, you need at least three polygons to meet at each vertex. Additionally, you need the sum of angles at each vertex to be less than 360°, because 360° is a flat plane, not a three-dimensional shape. This greatly limits the types of regular polyhedra that can be formed.

For example, if three squares meet at a single vertex, you may have part of a cube. If four squares meet at a single vertex, you would have a flat plane, not a solid. And, try as you might, you would never be able to connect five or more squares at a single vertex—it is simply impossible.

SPHERES

A sphere is a three-dimensional solid that is perfectly round. It is defined as the set of all points in space that are a given distance from its center. A sphere has no edges or vertices, and is not a polyhedron.

The distance from the center to the surface of a sphere is called the radius, and twice the radius is called the diameter.

A sphere is formed by rotating a circle (or semicircle) about its diameter.

The cross-section of a sphere is always a circle. Cross-section is the term for the intersection of a plane through a solid, which forms a two-dimensional shape. Imagine slicing through an orange—the flat surface of the cut piece would be an approximate circle. If you slice through a perfectly spherical solid, the cross section would always be a perfect circle (not an ellipse), no matter what angle you cut it from.

A great circle is the largest circle that can be drawn around a given sphere. If you make a cross section that passes through the sphere’s center, this forms a great circle. If a cross section does not pass through the sphere’s center, that’s known as a small circle.

True or False? A great circle has the same radius as the sphere it inscribes.

True! If you find a great circle of a sphere, the radius of the great circle will be the same as the radius of the sphere.

Imagine cutting a sphere exactly in half, through its center. The cross-section of the cut face will be a circle with the same radius as the sphere. This is also the largest possible circular cross-section of the sphere.

PRISMS

A prism is polyhedron that has a pair of congruent, parallel faces (called the bases of the prism). The bases are on opposite ends of the prism, and the other faces (sometimes called side faces) are always parallelograms.

A prism is named for the shape of its base—for example, “triangular prism” or “pentagonal prism.” In the figures below, the shaded parts show one base of each prism.

A prism is regular if its bases are regular polygons. Otherwise, the prism is irregular.

A prism is right if the bases are perpendicular to the other faces. Otherwise, it is an oblique prism. In a right prism, the side faces are always rectangles. All of the prisms in the above figure are right prisms.

Any cross-section parallel to the base will always be congruent to the base.

Other cross-sections will often form rectangles or parallelograms, but they may also form triangles, trapezoids, or other shapes. The number of sides in your cross-section is equal to the number of faces you slice through!

A cross-section of a cube is a polygon. How many sides could that polygon have? (Select all that apply.)

Reminder: A cube has six faces.

□ 2

□ 3

□ 4

□ 5

□ 6

□ 7

3, 4, 5, and 6 are correct. To get a three-sided cross section, you would intersect three faces of the cube. To get a four-sided cross section, you would intersect four faces of the cube. And so on. Remember, the number of sides in your cross-section is equal to the number of faces you slice through.

2 is not correct, because a polygon must have more than two sides! Moreover, if you “slice through” exactly two faces of a cube, you’d actually just be touching one of the edges, which gives you a line, not a polygon.

7 is not correct, because a cube has only six faces. For a seven-sided cross section, you would need a polyhedron with seven or more faces.

CYLINDERS

A cylinder is analogous to a prism, but its bases are circles. The radius of a cylinder is the radius of its circular base, and twice the radius is the diameter. The distance between the two bases is referred to as the height.

A cylinder is formed by rotating a rectangle about its edge, or about its center line.

Any cross-section parallel to the base will be a circle congruent to the base.

A cross-section perpendicular to the base will always be a rectangle.

Other, “slanted” cross-sections will be an ellipse, or a truncated ellipse (an ellipse with one or both ends cut off).

PYRAMIDS

A pyramid is a solid with a polygonal base, and triangular faces that meet at a vertex. This “top” vertex of the pyramid is known as the apex. The height or altitude of the pyramid is a line drawn from the apex perpendicular to the base. The slant height, conversely, is measured along a two-dimensional face.

A pyramid is named for the shape of its base—for example, “triangular pyramid” or “pentagonal pyramid.”

A pyramid is regular if its base is a regular polygon. Otherwise, the pyramid is irregular.

A pyramid is right if the apex is directly above the center of the base. Otherwise, it is an oblique pyramid.

In a right pyramid, any cross-section parallel to the base will be similar to the base, but smaller. The cross-section is not congruent to the base.

Other cross-sections may also form triangles, quadrilaterals, or other shapes.

CONES

A cone is analogous to a pyramid, but it has a circular base. Other than the base, a cone is considered to have one “side,” which is curved. The top point of a cone is called the apex.

A cone is formed by rotating a right triangle about one of its legs. The radius of the base of the cone would be equal to the other leg of the triangle.

Any cross-section parallel to the base will be a circle, which is smaller than the base.

Other cross-sections will be ellipses or parabolas.