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1. Three Dimensional Figures and Cross Sections
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Chapter 10
1. 

Three Dimensional Figures and Cross Sections

Understanding three-dimensional figures and their cross-sections is fundamental in geometry and its applications. These concepts allow for the visualization and analysis of shapes in real-world contexts, helping to identify their properties, relationships, and dimensions. Exploring elements such as solids, planes, and nets provides clarity on how these figures can be constructed, deconstructed, and visualized. Practical uses include designing objects, interpreting technical drawings, and solving spatial problems. By examining cross-sections of figures like prisms, cylinders, cones, and pyramids, insights into their structure and geometry are revealed. This understanding is key for mastering spatial reasoning and mathematical problem-solving, which are widely applicable across fields like architecture, engineering, and science.
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Lesson Settings & Tools
20 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Three Dimensional Figures and Cross Sections
Slide of 20
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 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.

Explore

Comparing the Cross-Sections of Two Figures

Consider two different 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.

A sphere, a cylinder, a pyramid
Three-dimensional is often written as so these figures are commonly called 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.

Net of a pentagonal prism

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.

Two different nets of a cube
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 geometric object. On the first level of the game, her character is hosting a party where her friends give her many cool presents.
Gifts 1-4 and wrappings 1-4
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

First, consider the first gift.

Gift 1

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

Wrapping 3
Therefore, Wrapping corresponds to Gift See how this wrapping can be folded into a cube.
A net of a cube folds into a cube

Gift

Now consider the shape of the Gift

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 matches this description!

Wrapping 4
Therefore, Wrapping is the wrapping of Gift See how it can be folded into a pyramid.
A net folds into a pyramid

Gift

This time consider the shape of the Gift

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.

Wrapping 2
Looking at all the given wrapping options, only Wrapping has a rectangle with two circles attached to the sides. This means that Gift corresponds to Wrapping
A net is folded into a cylinder

Gift

Finally, examine the shape of Gift

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
Wrapping consists of three rectangles and two triangles, so it matches the surface of Gift perfectly.
A net folds into a triangular prism
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.

Plane P, or plane ABC, contains the points A, B, and C.
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.

Coplanar points A, B, C, and D lie on a plane, while point E is situated above the plane and is non-coplanar.
In this illustration, and are coplanar, as they are all on the same plane. By contrast, is not on the same plane as the other points, so it is not coplanar with them.
Coplanar and non-coplanar points
Since a plane can be defined by any three points, 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 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:

  1. They can overlap. In that case, the planes are essentially the same plane and cannot be distinguished from each other.
  2. They can intersect. In that case, they share a common line.
  3. 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.
Two parallel planes
Parallel planes are denoted by using the symbol . For example, parallel planes and can be denoted as
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.

Different cross-sections of 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.

A polyhedron
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.
Polyhedron
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.
Polyhedron
Click on each vertex to paint it purple, then write out the list of vertices.
There are in total 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.
Polyhedron with painted vertices
Click on each edge to paint it pink, then write out the list of edges.
This polyhedron has a total of 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.
Polyhedron with painted edges
Click on each face to paint it in a darker shade of blue color, then write down a complete list of the faces.
There are 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.
Prism
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.
Oblique prism and right prism

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

Start by analyzing the cross-sections of Solid
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

Next, consider the cross-sections of Solid
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.
The first box with goods has been found!

Solid

This time examine the cross-sections of Solid
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

Lastly, analyze the cross-sections of Solid
Cross-sections of Solid 4
The cross-sections are parallelograms that could possibly be rectangles or even squares. They are all the same size. This suggests that the solid is a prism.
Notice that the cross-sections in this prism are not located strictly on top of each other like they were in previous three solids — instead, the cross-sections seem to move a little to the right as the laser moves up. This indicates that the solid is an oblique prism.
Discussion

Other Types of Solids

Not all solids are prisms. Some might have curved sides or not have two identical polygonal bases. This section will introduce a few nonprismatic solids.

Concept

Cylinder

A cylinder is a three-dimensional figure that has two circular bases that are parallel and equal in size, connected by a curved surface.

A cylinder both of the bases are depicted
The axis of a cylinder is the segment that connects the center of the bases. The height of a cylinder is the perpendicular distance between the bases. The radius of the cylinder is the radius of one of the bases.
Cylinder
If the axis of a cylinder is not perpendicular to the bases, it is called an oblique cylinder.
Oblique Cylinder
Discussion

Cone

A cone is a three-dimensional solid with a circular base and a point, called the vertex or apex, that is not in the same plane as the base. The altitude of a cone is the segment that runs perpendicularly from the vertex to the base.

A cone with a base radius r, depicting the vertex, altitude, and the base.
The length of the altitude is called the height of the cone. If the altitude intersects the base at the center, the cone is a right cone. In a right cone, the distance from the vertex to a point on the edge of the base is called the slant height of a cone.
Interactive cone where the height and radius can be change. The slant height is highlighted.
If a cone is not a right cone, it is called an oblique cone. Oblique cones do not have a uniform slant height.
Interactive Cone changing from oblique to right
Discussion

Pyramid

A pyramid is a polyhedron that has a base, which can be any polygon, and faces that are triangular and meet at a vertex called the apex. The triangular faces are called lateral faces. The altitude of a pyramid is the perpendicular segment that connects the apex to the base, similar to the altitude of a triangle.

Pyramid

The length of the altitude is the height of the pyramid. If a pyramid has a regular polygon as its base and congruent, isosceles triangles as its lateral faces, it is called a regular pyramid. The altitude of each lateral face in a regular pyramid is also known as the slant height of the pyramid.

Height and slant height of pyramid

If the apex of the pyramid is over the center of its base, it is called a right pyramid. Otherwise, it is called an oblique pyramid.

Oblique and right pyramids
Pop Quiz

Identifying Different Solids

Determine whether the given solid is a prism, a cylinder, a pyramid, or a cone.

A different solid is randomly generated
Closure

Space Occupied by a Solid

In this lesson, different three-dimensional figures were introduced. Consider the Venn diagram that illustrates the relations between some of the concepts covered this lesson.

A Venn diagram showing that polyhedra are a specific type of solid with flat faces. Solids can have flat faces or curved surfaces like spheres, cones, and cylinders. All solids are three-dimensional figures.

Additionally, different parts of solids such as vertices, edges, and faces, were named and identified. Vertices of a solid can be located using coordinates, edges can be measured by length, and the area of the faces can be calculated. However, what about the space inside a solid? Is there a way to measure it?

A prism with a question mark for the inside space
Yes, there is! The amount of space a solid occupies is known as its volume. It is possible to calculate the volume of different types of solids. This topic will be explored in greater detail later in the course.



Level 1
Level 2
Level 3
Three Dimensional Figures and Cross Sections
Exercises

Which of the following could be a net of a right rectangular prism that has a height of units and bases that are units long and units wide?

Four options

We are asked to draw a net of a rectangular prism. First, let's recall the definition of a prism.

Prism |-A prism is a polyhedron with two congruent polygon bases that lie in parallel planes. The remaining faces are parallelograms that have common sides with the bases.

A rectangular prism is a prism with rectangular bases. This means that a net of a rectangular prism will contain two identical rectangles for the bases of the prism. Since the bases are rectangles and the prism is said to be right, the prism has four rectangular side faces. Bases:& 2 rectangles Faces:& 4 rectangles We are also given the dimensions of the prism. It has a height of 5 units, so the length of side faces is 5 units. The bases are 3 units long and 2 units wide. Since the bases are connected to the side faces, we can conclude that some of the faces are 3 units wide and some 2 units wide. Bases:& 2 rectangles 3* 2 Faces:& 2 rectangles 5* 3 and & 2 rectangles 5* 2 We can use this information to draw a possible net of this right rectangular prism.

This corresponds to option A.

Identify the shape of each cross-section.

Cross-section-rect-prism1.png
Cross-section-cone1.png

We need to determine the shape of the horizontal cross-section of a rectangular prism. Notice that the cross section is parallel to the base of the prism, which is a rectangle. This indicates that the horizontal cross-section should also be a rectangle. Let's make sure by looking at the diagram from different perspectives.

We can see that the given cross section is indeed a rectangle.

This time we are given a cross section of a cone. We want to determine the shape of the cross section. Let's analyze the cross section by looking at the given diagram from different perspectives.

As we can see, the given cross section is a triangle.

There is a relationship between the number of edges, vertices, and faces of polyhedrons. Complete the table and describe the pattern in the table.

Polyhedron Faces Vertices Edges
Triangular Pyramid
Rectangular Pyramid
Triangular Prism
Rectangular Prism

We need to find the number of faces, vertices, and edges there are in four different polyhedrons. Let's analyze each polyhedron one at a time.

Triangular Pyramid

Let's start with the triangular pyramid. This is a polyhedron in which the base is a triangle and the lateral faces are triangles that meet at the apex of the pyramid.

Let's count the number of vertices, edges, and faces it has. We can see that there are 4 vertices, 6 edges, and 4 faces.

Polyhedron Faces (F) Vertices (V) F+V Edges (E)
Triangular Pyramid 4 4 4+4=8 6
Rectangular Pyramid
Triangular Prism
Rectangular Prism

Rectangular Pyramid

Now we will consider a rectangular pyramid. This is a pyramid whose base is a rectangle.

This pyramid has 5 vertices, 8 edges, and 5 faces. Let's fill in these values in the table.

Polyhedron Faces (F) Vertices (V) F+V Edges (E)
Triangular Pyramid 4 4 4+4=8 6
Rectangular Pyramid 5 5 5+5=10 8
Triangular Prism
Rectangular Prism

Triangular Prism

Next, let's analyze a triangular prism. This is a polyhedron with triangular bases connected by lateral faces that are parallelograms or rectangles.

We can see that a triangular prism has 6 vertices, 9 edges, and 5 faces.

Polyhedron Faces (F) Vertices (V) F+V Edges (E)
Triangular Pyramid 4 4 4+4=8 6
Rectangular Pyramid 5 5 5+5=10 8
Triangular Prism 5 6 5+6=11 9
Rectangular Prism

Rectangular Prism

The last solid we need to examine is a rectangular prism. This is a prism whose bases are rectangles. Let's do it!

This polyhedron has 8 vertices, 12 edges, and 6 faces. Let's write these values in the table to complete it.

Polyhedron Faces (F) Vertices (V) F+V Edges (E)
Triangular Pyramid 4 4 4+4=8 6
Rectangular Pyramid 5 5 5+5=10 8
Triangular Prism 5 6 5+6=11 9
Rectangular Prism 6 8 6+8=14 12

Conclusion

Finally, let's analyze the whole table and try to find a pattern. We will take a close look at F+V and E.

Polyhedron Faces (F) Vertices (V) F+V Edges (E)
Triangular Pyramid 4 4 4+4=8 6
Rectangular Pyramid 5 5 5+5=10 8
Triangular Prism 5 6 5+6=11 9
Rectangular Prism 6 8 6+8=14 12

Notice that the sum of the number of faces and vertices is always 2 more than the number of edges. This means the following pattern must be true. F+V-2=E

Determine whether the given statement is always, sometimes, or never true.

A prism has bases and faces.

We are asked to determine whether the following statement is always, sometimes, or never true.

A prism has 2 bases and 5 faces.

Let's start by recalling that a prism is a three-dimensional figure with at least two parallel, congruent faces, called bases, that are polygons. We can draw a few example prisms and analyze them.

Notice that the prism on the left has 2 bases and 3 faces, the middle prism has 2 bases and 4 faces, and the prism on the right has 2 bases and 5 faces. This means that it is possible for a prism to have 2 and 5 faces, but it does not always have 2 bases and 5 faces. Therefore, the statement is sometimes true.

Three Dimensional Figures and Cross Sections

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