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