body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

Recommended

Tests

An error ocurred, try again later!

eCourses /

Pre-Algebra /

Chapter {{ article.chapter.number }}

{{ article.number }}.

Show less Show more

Lesson Settings & Tools

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}

Image Credits

{{ item.file.title }}
{{ presentation }}

No file copyrights entries found

People live in a three-dimensional world and are surrounded by many three-dimensional figures. This lesson will explore some of the different types of

3 D

figures that are commonly studied in math and analyze their cross-sections.

Catch-Up and Review

Here is a recommended reading before getting started with this lesson.

Two-Dimensional Figures
Polygon

Explore

Comparing the Cross-Sections of Two $3 D$ Figures

Consider two different

3 D

figures. A plane parallel to their bases passes through both of them, creating their cross-sections.

A plane that runs parallel to the bases of a cone and cylinder, creating cross-sections of each.

What conclusions can be made about the cross-sections of Figure I? Are the cross-sections all the same or different? What about the cross-sections of Figure II?

Discussion

Three-Dimensional Figures

Most objects in the real world have three dimensions. Consider the definition of a three-dimensional figure.

Concept

Three-Dimensional Figure

A three-dimensional figure is a geometrical figure that has three dimensions — length, width, and height. Unlike two-dimensional figures, three-dimensional figures have height, which can also be referred to as thickness or depth.

A rectangular prism with the length, width, and height identified

All three-dimensional figures occupy space, which is measured in terms of volume. Some examples of basic three-dimensional figures are spheres, cones, pyramids, cubes, and prisms.

Three-dimensional is often written as

3 D,

so these figures are commonly called

3 D

figures.

Discussion

Solid

A solid is a stable three-dimensional object with its interior completely filled. An object containing a fluid, for example, is not a solid. Solids are completely enclosed, occupy space, and have a definite shape and volume. Solids can have flat faces, curved surfaces, or a combination of both. Here are some examples of solids.

A cylinder, a pyramid, a sphere, rocks, a nickel, golden bars

External credits: obv_design, Wikipedia, user2122532

The surface of a solid can be examined by creating a net of the solid.

Discussion

Net of a Solid

A net of a solid is a two-dimensional representation of a three-dimensional figure. A net shows all the faces of a figure in one view and can be folded into the three-dimensional shape.

Note that the net of a solid is not unique, as a three-dimensional figure can have multiple nets. However, the area of each net of a solid is always the same and is equal to the surface area of the solid.

Example

Matching Solids With Their Nets

Dominika is playtesting a new video game that her friend made. Every single thing in the game is a

3 D

geometric object. On the first level of the game, her character is hosting a party where her friends give her many cool presents.

Dominika is very excited to open all the presents! Only later she realizes that the names of gift-givers are written on the wrapping paper of the presents. Dominika is asked to figure out who gave her which gift and send thank you cards, so she needs to match the wrapping paper to the gifts.

Hint

Analyze the surface of each gift. Do their sides resemble any geometric figures such as squares, rectangles, triangles, or circles? Find the wrapping paper that consists of the same figures.

Solution

To match the gifts to their wrapping paper, the shape of each gift will be analyzed. Notice that wrappings match the shape of the surface of each gift. This means that the wrapping papers are basically the nets of the solids.

Gift $1$

First, consider the first gift.

It has the shape of a cube. Including the sides of the box not visible from this angle, there are $6$ square sides. Therefore, its net will be made up of exactly $6$ squares that can be folded into a cube. Now consider the given wrapping papers. Wrapping $3$ consists of exactly $6$ squares, so it matches the sides of the gift box perfectly.

Therefore, Wrapping

3

corresponds to Gift

1 .

See how this wrapping can be folded into a cube.

Gift $2$

Now consider the shape of the Gift $2 .$

This gift box has four triangular sides and a rectangular or square base. This means that the matching wrapping should consist of a rectangle or a square and four triangles attached in a way that allows it to be folded into a pyramid. Notice that Wrapping $4$ matches this description!

Therefore, Wrapping

4

is the wrapping of Gift

2 .

See how it can be folded into a pyramid.

Gift $3$

This time consider the shape of the Gift $3 .$

This gift box has two circular parallel bases joined by a curved side surface. The curved surface could be covered by a long rectangular piece of wrapping paper. To cover the circular bases, there should be two circles attached to the longer sides of the rectangle. In other words, its net would be a rectangle with two circles attached.

Looking at all the given wrapping options, only Wrapping

2

has a rectangle with two circles attached to the sides. This means that Gift

3

corresponds to Wrapping

2 .

Gift $4$

Finally, examine the shape of Gift $4 .$

It has two triangular bases at the top and bottom connected by three rectangular lateral sides. Look for a net that consists of these parts to find the match.

Wrapping

1

consists of three rectangles and two triangles, so it matches the surface of Gift

4

perfectly.

Discussion

Definition of a Plane

Visualize a flat surface in a three-dimensional environment, such as a tabletop or a sheet of paper. How can this object be described?

Concept

Plane

A plane is a two-dimensional object that has infinite width and length but no height. An endless sheet of paper can represent a plane. A plane can be named by using three points on it or by using a capital letter.

Exactly one plane can pass through any three non-collinear points.

A certain relationship exists between points defined by their position on the same or different planes.

Discussion

Coplanar Points

Any set of points that lie on the same plane are said to be coplanar.

In this illustration,

A,

B,

C,

and

D

are coplanar, as they are all on the same plane. By contrast,

E

is not on the same plane as the other points, so it is not coplanar with them.

Since a plane can be defined by any three points,

E

is still coplanar with other pairs of points. Therefore, there are seven sets of coplanar points on the graph.

Pop Quiz

Points and Planes

Consider a plane and some points located in a $3 D$ space. Determine which points are coplanar, which points do not lie on the plane, or what a possible name for the plane could be. Note that the plane can be viewed from different perspectives by clicking and dragging the mouse.

A plane and some points are randomly generated

Discussion

Positions of Two Planes

Two planes can have three positions in respect to each other:

They can overlap. In that case, the planes are essentially the same plane and cannot be distinguished from each other.
They can intersect. In that case, they share a common line.
They can be parallel.

Here is a deeper dive into the last case.

Concept

Parallel Planes

Two distinct planes are called parallel if they never intersect each other, similar to parallel lines. Parallel planes do not share any common lines or points.

Parallel planes are denoted by using the symbol

∥

. For example, parallel planes

P

and

M

can be denoted as

P ∣ ∣ M .

Discussion

Cross-Sections of Solids

A plane can be used to examine the interior of a solid. The process involves drawing a plane through the solid, which results in a certain shape called a cross-section.

Concept

Cross-Section

The intersection between a plane and a solid creates a two-dimensional shape known as a cross-section. The cross-section produced depends on where the plane intersects the solid. Here are some examples of cross-sections that can be created from a sphere, a cone, and a cylinder.

Discussion

Polyhedron

A polyhedron is a three-dimensional figure whose surfaces are polygons. Each of these polygons is a face of the polyhedron. An edge is a segment formed by the intersection of two faces. A vertex is a point where three or more edges meet.

Example

Coloring the Polyhedron by Parts

On the next level, Dominika sees a huge polyhedron with an interesting shape. She is asked to paint its parts in different colors: the vertices in purple, the edges in pink, and the faces in a darker shade of blue.

Complete this task along with Dominika! Write the names of all the vertices, edges, and faces while painting them in the needed colors. How many vertices, edges, and faces does this polyhedron have?

Hint

Recall the definitions of the vertices, edges, and faces of a polyhedron.

Solution

It is given that Dominika needs to paint different parts of the polyhedron in different colors. First, recall the definitions of a face, an edge, and a vertex of a polyhedron.

A face is a flat surface of a polyhedron in the shape of a polygon. An edge is a segment formed by the intersection of two faces. A vertex is point where three or more edges intersect.

Start by identifying the vertices of the polyhedron. These are the points where the edges of the polyhedron meet and they are labeled with capital letters.

Click on each vertex to paint it purple, then write out the list of vertices.

\underline{Vertices :} A, B, C, D, E, F, G, H, I, J, K, L

There are in total

12

vertices in this polyhedron. Next, consider the edges of the polyhedron. These are the segments between each pair of vertices and are often named using the two capital letters of the vertices.

Click on each edge to paint it pink, then write out the list of edges.

\underline{Edges :} A B, B F, C H, E I, K J, H L, A C, B K, C D, E J, K L, H I, A D, B G, D H, E F, K G, I L, A E, B C, D I, F J, G L, I J, A F, C G, D E, F K, G H, L J

This polyhedron has a total of

30

edges. Finally, consider the faces of the polyhedron. These are the polygons that make up the surface of the polyhedron. Each face can be named by the vertices that make it up.

Click on each face to paint it in a darker shade of blue color, then write down a complete list of the faces.

A B C, B C G, E J I, G K L, A C D, C H G, E F J, G H L, \underline{Faces :} A D E, C D H, F K J, H I L, A E F, D H I, B F K, J I L, A B F, D E I, B G K, J K L

There are

20

faces in total. Dominika has successfully painted the whole polyhedron in its new colors!

Discussion

Prisms and Their Cross-Sections

An often used type of solid that can be found in various forms in the real world is the prism.

Concept

Prism

A prism is a three-dimensional object created by connecting a polygon with a translated version of the same polygon, vertex to vertex. The two parallel congruent polygons are called bases. The other faces are called lateral faces. The intersection of two lateral faces is called a lateral edge.

A prism with perpendicular lateral faces and bases is a right prism. Otherwise, it is is called an oblique prism. The lateral faces of a prism can either be rectangles or parallelograms. In an oblique prism, at least one lateral face must be a parallelogram. In contrast, in a right prism, all lateral faces are rectangles.

When a plane is drawn through a prism, it creates a cross-section of the prism. It is important to note that all cross-sections that are parallel to a base of a prism are identical to each other and to the base.

A plane intersects a hexagonal prism parallel to its bases forming the same cross-sections

Example

Identifying Prisms

Dominika successfully painted the polyhedron and beat the level in the game. On the next level, she appears in a completely dark room. Different solids are placed in the middle of the room. Dominika needs to identify the prisms because they have things inside of them needed for the next levels.

Images of the interior region of four different solids scanned by the laser in a dark room

She can only use a laser that x-rays the entire room parallel to the floor and makes the outline of the solid visible. Which solids should she identify as prisms?

Hint

Recall the definition of a prism. What geometric shapes can the bases and lateral faces of a prism have? Use the fact that all cross-sections of a prism parallel to the base are the same.

Solution

Dominika is asked to identify which solids are prisms. First, recall the definition of a prism.

Definition of a Prism
A prism is a polyhedron where two faces, called bases, are congruent polygons lying in parallel planes, and the remaining faces are parallelograms that have common sides with these polygons.

Notice that the laser scans the room and each solid parallel to the floor and, therefore, to the base of the solid. This means that the laser shows cross-sections of solids parallel to their bases. Since Dominika needs to identify all the prisms, she can use the following fact to guide her.

All cross-sections of a prism parallel to a base are identical to each other and to the base.

Now, examine the cross-sections of each given solid one at a time.

Solid $1$

Start by analyzing the cross-sections of Solid

1 .

All the cross-sections are pentagons of different sizes. The cross-section at the bottom of the solid has the greatest area. As the laser moves up, the pentagons get smaller and smaller. The last visible cross-section seems to be a point. Since the cross-sections are not identical, this solid is not a prism.

Solid 1 : Not a prism \times

Solid $2$

Next, consider the cross-sections of Solid

2 .

All the cross-sections are hexagons of the same size. This means that the solid is a prism with hexagonal bases.

Solid 2 : A prism ✓

The first box with goods has been found!

Solid $3$

This time examine the cross-sections of Solid

3 .

The cross-sections are ellipses that might be circles. They all have the same size, which fulfills the requirement about the cross-sections being identical. However, the lateral faces of prisms are parallelograms and the bases are polygons. This solid has a smooth, curved lateral surface and the bases are ellipses, which are not polygons. Therefore, this is not a prism.

Solid 3 : Not a prism \times

{{ article.displayTitle }}

Catch-Up and Review

Hint

Solution

Gift 1

Gift 2

Gift 3

Gift 4

Hint

Solution

Hint

Solution

Solid 1

Solid 2

Solid 3

Gift $1$

Gift $2$

Gift $3$

Gift $4$

Solid $1$

Solid $2$

Solid $3$