POLYHEDRONS
In this unit, we will discuss three dimensional (3D) figures. A 3D figure made up of polygons is called a polyhedron. The polygons are called the faces of the polyhedron. Each side of the face is called an edge. The point where three or more edges meet is called a vertex. The example below is a polyhedron because each face is either a pentagon or a hexagon (both polygons). |
PRISMS
Suppose you buy a fresh loaf of bread from a bakery, but it isn’t fully sliced! The store worker offers to put it through the slicer for you and asks which direction you would like it sliced. You want to make it so that everybody will have the same sized slice. How would you direct the store worker to slice the bread? (Recall: a bread slicer cuts parallel slices!). The above example helps us determine whether or not a polyhedron is a prism. If you can make parallel slices where everybody will have the same size and shape, then the shape can be called a prism, and the face of each slice is actually the prism’s base. In other words, you can slice the loaf of bread into parallel slices that are all congruent. In math terms, a prism is a polyhedron that has two congruent polygons that are parallel to one another (called the bases), separated by an equal distance (called the height of the prism). Thus all the other faces must be parallelograms, or rectangles for right prisms. Each prism is named by its base. The example below is called a pentagonal prism, since its bases are each pentagons: |
We will label the bases and height of the prism. Notice that the bases are parallel to one another.
If you imagine unfolding these polyhedrons, you will have a net. Click on the figure below to see different polyhedron and their nets. You can also find how many edges, vertices, and faces that each of the figures have.
Below is an example of a net of the pentagonal prism. Note that it must have two congruent pentagons for bases and the rest of the faces, which are lateral faces, are rectangles.
Example 1:
Are the following figures prisms?
If the solid is a prism, classify it and sketch the net, including the label for the height of the prism. If the solid is not a prism, explain why not.
Are the following figures prisms?
If the solid is a prism, classify it and sketch the net, including the label for the height of the prism. If the solid is not a prism, explain why not.
1a) Yes, this is a triangular prism (though you cannot see the other triangle base, it is in the background! Also notice that the other lateral faces are rectangles:
b) No, this is not a prism. There are not two congruent, parallel polygons for its bases. Also, its faces are not rectangles!
c) No, this is not a prism. Notice that there aren'tt two congruent bases that are separated by rectangles.
d) Yes, this is a hexagonal prism. There are two congruent hexagons that are parallel to each other and separated by congruent rectangular faces.
c) No, this is not a prism. Notice that there aren'tt two congruent bases that are separated by rectangles.
d) Yes, this is a hexagonal prism. There are two congruent hexagons that are parallel to each other and separated by congruent rectangular faces.
CYLINDERS
Cylinders are much like prisms, but their bases are circular. Right cylinders consist of two congruent circular bases and a rectangular lateral face that connects them. The height of a cylinder is the perpendicular distance between the circle bases.
Cylinders are much like prisms, but their bases are circular. Right cylinders consist of two congruent circular bases and a rectangular lateral face that connects them. The height of a cylinder is the perpendicular distance between the circle bases.
PYRAMIDS
A pyramid is a polyhedron with a single polygon base and triangular lateral faces that meet at one point called a vertex. Each pyramid is named by its base, and the height of a pyramid is the perpendicular distance from the vertex to the plane of the base. A regular right pyramid is one whose base is a regular polygon and the triangular faces are congruent isosceles triangles. Take a look at the examples below:
A pyramid is a polyhedron with a single polygon base and triangular lateral faces that meet at one point called a vertex. Each pyramid is named by its base, and the height of a pyramid is the perpendicular distance from the vertex to the plane of the base. A regular right pyramid is one whose base is a regular polygon and the triangular faces are congruent isosceles triangles. Take a look at the examples below:
The net of a pyramid will consist of the base and then triangles for its lateral faces. The slant height of the pyramid, noted with a cursive lower case \(\ell\), is the height of one of the triangular faces. Let’s look at the square pyramid to investigate the slant height:
Here is a net of the square pyramid. Notice that is has one square for its base, and four triangles that make up the lateral faces.
Example 2:
Here is a net of the square pyramid. Notice that is has one square for its base, and four triangles that make up the lateral faces. a) Name the height of the pyramid. b) Name a lateral face of the pyramid. c) Name the base of the pyramid. d) Name the slant height of the pyramid. Solution: a) \(\overline{HJ}\) b) \(\triangle{HAG}\), \(\triangle{HBA}\), \(\triangle{HCB}\), \(\triangle{HCD}\), \(\triangle{HFD}\) or \(\triangle{HGF}\) c) hexagon \(ABCDFG\) |
Example 3:
The following is a regular right pyramid. If the perimeter of its base is \(28\) inches and its slant height is \(15\) inches, determine the length of the height of the pyramid.
The following is a regular right pyramid. If the perimeter of its base is \(28\) inches and its slant height is \(15\) inches, determine the length of the height of the pyramid.
Solution:
Since this is a regular pyramid, the base must be a square. So the length of each side of the base must be \(7\) inches in order for the perimeter of the base to be \(28\) inches. Let’s label the diagram to show the information given:
Since this is a regular pyramid, the base must be a square. So the length of each side of the base must be \(7\) inches in order for the perimeter of the base to be \(28\) inches. Let’s label the diagram to show the information given:
Recall that the faces of this pyramid must be isosceles triangles because this is a regular right pyramid. Let’s focus on the right triangle highlighted below:
Let’s use the Pythagorean theorem to solve for the missing height:
\(\begin{align*}\\
3.5^2 + h^2 &= 15^2\\
12.25 + h^2 &= 225\\
h^2 &= 212.75\\
h &= \pm\sqrt{212.75}\\
h &= 14.585921...
\end{align*}\)
We use the positive value because length is positive
\(h \approx 14.6\) in
CONES
Cones are like a pyramids with a circular base. The height of a cone is a segment perpendicular to the base that goes from the center of of the circle to the vertex of the cone. The slant height \(\ell\) is any segment that can be drawn from point on the circle to the apex. Take a look at two different slant heights drawn below:
\(\begin{align*}\\
3.5^2 + h^2 &= 15^2\\
12.25 + h^2 &= 225\\
h^2 &= 212.75\\
h &= \pm\sqrt{212.75}\\
h &= 14.585921...
\end{align*}\)
We use the positive value because length is positive
\(h \approx 14.6\) in
CONES
Cones are like a pyramids with a circular base. The height of a cone is a segment perpendicular to the base that goes from the center of of the circle to the vertex of the cone. The slant height \(\ell\) is any segment that can be drawn from point on the circle to the apex. Take a look at two different slant heights drawn below:
The net of a cone consists of a circle (which would be the base) and a sector (which is the lateral area):
Example 4:
Find the length of the slant height of a cone whose diameter is \(10\) cm and height is \(12\) cm
Solution:
Let’s sketch a picture of the cone described. Watch the video for the solution.
Find the length of the slant height of a cone whose diameter is \(10\) cm and height is \(12\) cm
Solution:
Let’s sketch a picture of the cone described. Watch the video for the solution.
OBLIQUE and RIGHT 3-D FIGURES
An oblique figure is one that is distorted so that is seems to 'lean over' at an angle, as opposed to being perpendicular to the base. Figures that are not oblique are right. See the right prism and oblique prism below.
An oblique figure is one that is distorted so that is seems to 'lean over' at an angle, as opposed to being perpendicular to the base. Figures that are not oblique are right. See the right prism and oblique prism below.
SPHERES AND HEMISPHERES
A sphere is a three dimensional circle. Unlike the pyramid and unlike the prism, it actually has no base. The radius is a segment that goes from the center of the sphere to a point on the sphere. Use the following to move around the point or rotate the sphere. See how many radii you can create:
A sphere is a three dimensional circle. Unlike the pyramid and unlike the prism, it actually has no base. The radius is a segment that goes from the center of the sphere to a point on the sphere. Use the following to move around the point or rotate the sphere. See how many radii you can create:
Quick Check
1) Classify the following figure by the net shown. Then, label its parts.
1) Classify the following figure by the net shown. Then, label its parts.
2) The following problem asks to label the slant height of the pyramid below.
Jazzy’s answer is below. Describe and correct the error made in the Jazzy’s answer to the problem.
3) Find the length of the height for a cone that would be constructed from the net below.