About Three Dimensional Shapes
- Understanding Three-dimensional Figures
- What are the faces, edges, and vertices of three-dimensional figures?
- The number of faces, edges, and vertices of some solids
- How to draw the front, the side, and the top views of a three-dimensional solid?
- Drawing three-dimensional figures (prisms, pyramids, and so on)
- Solved Examples
- Frequently Asked Questions
Understanding Three dimensional Figures
A three-dimensional figure or object is a solid that occupies space in our physical environment. Every three-dimensional figure has length, width, and height. Unlike two-dimensional objects, three-dimensional figures have an additional property: their height is also known as depth. We can notice various three-dimensional objects in our surroundings, like notebooks, pencils, cups, dice, ice cubes, cakes, and so on. Every object that we can hold in our hand is a three-dimensional object.
In mathematics, there are two types of geometry, First, plane geometry, that deals with two-dimensional figures. The geometrical shapes that are drawn on paper fall under plane geometry. Points, lines, triangles, rectangles, circles, and so on. are two-dimensional shapes or figures.
Second, solid geometry, which deals with three-dimensional figures. Cubes, cylinders, cuboids, spheres, and so on are a few examples of three-dimensional figures.
What are the faces, edges, and vertices of Three-dimensional figures?
A solid is a three-dimensional figure that encloses space. A polyhedron is also a solid whose faces are all polygons. So, we take an example of a polyhedron to understand the faces, edges, and vertices of a three-dimensional solid.
Face: The flat surface of a polyhedron is known as its face. There are two bases and four lateral faces of a polyhedron. So, there are a total of six faces in a polyhedron.
Edge: An edge is a line segment where two faces intersect. Each base has four edges, and the lateral surfaces form four perpendicular edges. So, there are a total of 12 edges in a polyhedron.
Vertex: A vertex is a point where three or more edges intersect. There are 8 vertices in a polyhedron.
The number faces, edges, and vertices of some solids
How to draw the front, the side, and the top views of a Three-dimensional solid?
When we look at a solid from a particular direction, like from the front, the top or the side, we will see a distinct view of the solid. These views are the front view, the top view, and the side view. Each of these views can be drawn on paper as a two-dimensional drawing. If we take a solid comprising three cubes as shown in the figure,
The front view is the face that we see from the front side as:
Similarly, for the side view, we see from the sideways, which appears as:
And the top view is the view from the upper side of the solid:
Drawing three-dimensional figures (prisms, pyramids, and so on)
How can we draw three-dimensional figures?
We can easily draw two-dimensional figures on paper using a scale, pencils, a compass, and set squares. However, drawing three-dimensional figures on paper is quite different. We draw lines and curves on paper such that they can easily be visualised as three-dimensional solids.
How can we draw a rectangular prism?
The following are the steps to draw a rectangular prism on paper:
Step 1: Draw two identical rectangular faces as shown in the figure.
Step 2: Connect the corresponding vertices using straight lines as shown.
Step 3: Draw the edges that are hidden or are on the backward side as dashed lines.
Finally, the three-dimensional rectangular prism is obtained.
How do we draw a triangular pyramid?
The following are the steps to draw a triangular pyramid on paper:
Step 1: Draw a triangle and mark a point above the triangle as shown in the figure.
Step 2: Connect all the three vertices of the triangle to the point.
Step 3: Mark any hidden edges as dashed lines.
Finally, we obtain the three-dimensional triangular pyramid.
Solved Examples
Example 1:
Find the numbers of faces, edges, and vertices in the figure given below.
Solution:
The figure has one pentagonal base and 5 triangular lateral faces.
The faces intersect at 10 different line segments.
The edges intersect at 6 different points.
So, the given figure has 6 faces, 10 edges, and 6 vertices.
Example 2:
Draw the front, the side and the top views of the solid shape given below.
Solution:
The front view is:
The side view is:
The top views is:
Example 3: Draw the top, the side and the front views of the solid cylinder.
Solution:
The front and the side views of the solid cylinder are both the same and are rectangular in shape.
The top view of a solid cylinder is in the form of a circle.
Example 4: Find the number of faces, edges, and vertices of the solid figure given below:
Solution:
The solid has 2 pentagonal faces, on the top and bottom. There are 5 lateral faces.
The faces intersect at 15 different line segments.
The edges intersect at 10 different points.
Therefore, the solid has 7 faces, 15 edges, and 10 vertices.
The solid shapes are three-dimensional objects having length, width, and height. Books, pencils, toys, erasers, bottles, and so on are a few examples of solids we see in daily life. Spheres, polyhedrons, cylinders, and cones are special geometrical solids.
A sphere is a type of three dimensional solid that does not have any vertex or edge