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| 20 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here is a recommended reading before getting started with this lesson.
Most objects in the real world have three dimensions. Consider the definition of a 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.
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-dimensionalis often written as
3D,so these figures are commonly called 3D figures.
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 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.
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.
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.
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.
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!
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.
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.
Visualize a flat surface in a three-dimensional environment, such as a tabletop or a sheet of paper. How can this object be described?
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.
Any set of points that lie on the same plane are said to be coplanar.
Two planes can have three positions in respect to each other:
Here is a deeper dive into the last case.
∥.For example, parallel planes P and M can be denoted as P ∣∣ M.
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.
Recall the definitions of the vertices, edges, and faces of a polyhedron.
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 often used type of solid that can be found in various forms in the real world is the 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.
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.
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.
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.
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 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 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.
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.
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.
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.
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?
Classify the solid.
We are asked to classify the following solid.
The surface of this solid is made up of polygons, which indicates that this is a polyhedron. Polyhedrons can be classified based on the following characteristics.
Notice that the given polyhedron has triangular sides and only one base. This suggests that the given polyhedron is a pyramid.
Also, we can see that the base of the pyramid is a pentagon. This makes it a pentagonal pyramid.
Let's consider the given solid.
Its surface is also made up of polygons, which means that this is a polyhedron. To classify this polyhedron, let's find the number of its bases and examine the shape of the bases and sides. We can see that the solid has two bases and three rectangular faces.
This suggests that the given polyhedron is a prism that lies on a side face instead of the base. We can also see that the bases of the prism are triangles. This makes it a triangular prism.
Identify the solid from its net.
We are asked to identify the solid based on its net.
This net shows four identical triangular faces. The triangular face in the middle could be the base and the other faces could be the sides of the figure.
This figure has one base and its sides are triangular. This means that it is a pyramid. What is more, since the base is a triangle, it is a triangular pyramid.
Let's begin by analyzing the shapes of the net parts.
We can see that the net consists of two parts, a circle, which can be the base of the solid, and a figure similar to a triangle with one curved side. It can represent the curved side surface of a cone.
Moreover, because the side surface is symmetrical, the top vertex must be located over the center of the base. This indicates that the cone is a right cone.
Let's start by recalling the definition of a rectangular prism.
Rectangular 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 a rectangular base.
Consider an example of a rectangular prism.
Let's draw a net of this prism.
We can see that there are 6 rectangles in the net of a rectangular prism. This means that Ramsha will have to draw 6 rectangles in her net.
We are asked to name all the coplanar points on the diagram. Let's start by recalling that coplanar points are points that lie on the same plane. There is one plane shown in the diagram. Rotating the diagram to view it from different perspectives can help us identify all the points that lie on the plane.
We can see that points C, E, and F lie on the same plane, while A, B, and D do not lie on that plane. Therefore, C, E, and F are coplanar points.
From Part A, we know that three points lie on the plane.
A plane can be named by any three points that lie on the plane. Let's write all the possible names of the plane using the points that lie in it. CEF, CFE, ECF, EFC, FEC, FCE Alternatively, we can name the plane by assigning an arbitrary capital letter to the plane. For example, P or S could also be names of the plane.
Identify the figure. Then name the bases, faces, edges, and vertices. What is the sum of the numbers of all the faces, edges, and vertices?
We are asked to identify the following polyhedron.
We can see that the polyhedron has two bases, ABC and EDF, and three rectangular side faces, EBCF, FCAD, and ABED. Bases: & ABC, EDF Faces: & EBCF, FCAD, ABED The bases are identical and the sides are polygons, which indicates that the polyhedron is a prism. Additionally, the bases are triangles, so this is a triangular prism. Next, write the names of all the edges by listing the segments where the faces intersect. Edges: AD, BE, CF, AB, BC, CA, DE, EF, FD The polyhedron has 9 edges. Now let's list all the vertices of the solid. Vertices: A, B, C, D, E, F We found that there are 6 vertices. Finally, we can add the numbers of all the faces, edges, and vertices. Note that the bases are also faces of the solid, so we will count them, too. 2+ 3+ 9+ 6=20
We are asked to identify the following polyhedron.
We can see that the polyhedron has 1 base, KLMNOP, and 6 triangular side faces. Bases: & KLMNOP Faces: & KLS, LMS, MNS, & NOS, OPS, PKS This indicates that the polyhedron is a pyramid. What is more, the base is a hexagon, so this is a hexagonal pyramid. Next, let's write the names of all the edges by listing the segments where the faces intersect. Edges: KL, LM, MN, NO, OP, PK, SK, SL, SM, SN, SO, SP The polyhedron has 12 edges. Let's now list all the vertices of the solid. Vertices: K, L, M, N, O, P, S We found that there are 7 vertices. Finally, let's count how many faces, edges, and vertices the polyhedron has in total. Note that the base is also a face of the solid, so we will count it, too! 1+ 6+ 12+ 7=26