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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.
Therefore, Wrapping $3$ corresponds to Gift $1.$ See how this wrapping can be folded into a cube.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.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 $2.$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.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.
Exactly one plane can pass through any three non-collinear 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.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.
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.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.
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.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?
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.