Geometric Solids: Properties, Formulas & Calculation

Imagine you are holding a shoe box. The shoebox is a geometric body. But what makes the shoebox a geometric body?

Geometric body – definition

You can touch the shoe box with your hands. But that is not enough for it to be a geometric body. The important thing is that the shoebox has contents. No shoes have to be in the box for this. It is sufficient that air can be in the shoe box. Mathematically, you can say, «The shoebox is three dimensional«.

A geometric body is a three-dimensional structure. It has volume. The geometric body is bounded by surfaces.

The surfaces delimiting the body can be flat or curved.

You can also pick up a sheet of paper and touch it. However, it is not a geometric body because it has no content and is therefore not three-dimensional.

A juice packet, on the other hand, is a geometric solid. It’s three dimensional. Its content is mostly juice.

But a book, for example, can also be a geometric body.

Properties of geometric bodies

The properties of a geometric solid are summarized:

  • The geometric body is three-dimensional.

  • The geometric body is bounded by surfaces.

  • The geometric body has edges (except the sphere).

  • The geometric body has corners. Some have a tip.

You can see the edges, surfaces, corners and the tip of a geometric body in Figure 1.

Figure 1: Faces, corners and edges of a geometric body

The individual surfaces of a body together form its surface. You can determine the size of the surface. It is called surface area.

The surface of a geometric body is based on the base or top surface and the lateral surface together. the Floor space is the «floor» of the geometric body that top surface the lid». The side surfaces form the mantle of the body.

The content of the geometric body becomes volume called. You can also calculate the volume of a geometric body.

Depending on the number and shape of the delimiting surfaces, corners and edges, different bodies are distinguished.

Geometric bodies – overview and names with formulas

Here you will find an overview of the most well-known geometric bodies with special names: cuboid, cube, cylinder, cone, pyramid, prism and sphere. If you click on the name of the body, an explanation will open with further information.

For every geometric body there is a formula with which you can determine the volume \(V\) of the body. \(G\) is always the base area of ​​the body and \(h\) is the height.

SurnameExplanationpictureformula volumecuboid

The shoe box in the example is a cuboid. It is bounded by six rectangles, with opposite rectangles being congruent.

Each cuboid has six faces, twelve edges and eight vertices.

Figure 2: Box

$$V=G h$$

or

$$V=l w h$$

with l length and b width

Dice

If all edges of a cuboid are the same length, it is one Dice.

Every cube is also a cuboid. That’s why everyone has dice six faces, twelve edges and eight vertices.

Figure 3: Cube

$$V=a^3$$with a length of an edge

A cylinder is a geometric body with congruent, parallel circles as base and top surface.

A cylinder consists of three faces and two edges. He has no corners. The lateral surface is curved.

Figure 4: Cylinder

$$V=G h$$$$G=\pi r^2$$cone

A cone consists of a circular base and a tip.

The cone has two faces and one edge.

Figure 5: Cone

$$V=\frac{1}{3} G h$$$$G=\pi r^2$$

A geometric body with an n-gon as the base, triangles as the sides and a point is called pyramid.

The number of corners, edges and faces of a pyramid depends on the base area.

Figure 6: Pyramid

$$V=\frac{1}{3} G h$$

A prism has congruent, parallel bottom and top surfaces. The side faces are rectangles.

Blocks, cubes and cylinders are also prisms.

Prisms can look very different, depending on the number and shape of corners, edges and faces.

Figure 7: Prism

$$V=G h$$Bullet

One Bullet is a completely round geometric body. All points on the sphere are the same distance from the center.

Figure 8: Sphere

$$V=\frac{4}{3}\pi r^3$$

Don’t forget to open the explanations for the individual geometric bodies by clicking if you want to know more.

composite bodies

There are also bodies that are composed of several geometric bodies.

composite bodies are two or more geometric bodies that are connected to each other. These bodies form a whole body.

Putting the bodies together changes the shapes. The number of corners, edges and surfaces is now also different.

In Figure 9, a cylinder and a sphere became a composite body.

Figure 9: Assembled body

See the Composite Bodies explanation if you want to know more.

Geometric solids – representations and drawing

Surely you have already tried to draw a geometric body on a piece of paper. The geometric body is three-dimensional. You can embrace him. The sheet of paper, on the other hand, is only two-dimensional. It has no volume. Now the trick is to make the body look like it’s three dimensional on the paper.

oblique image

One way to represent a geometric body is the oblique image.

A oblique image of a geometric solid is a three-dimensional representation of the body on a flat, two-dimensional surface. The front view remains unchanged, while the side and top surfaces are drawn foreshortened.

The images for the individual bodies in the table are oblique images.

In the «Oblique» explanation you will find a detailed description and instructions for drawing.

body mesh

You can also draw a body mesh from a geometric body. To do this, imagine that you cut open the geometric body and unfold it. In Figure 9 you can see a body mesh of a cube, also called a cube mesh. In the body mesh of a geometric body you can see the shapes of the surfaces.

Figure 10: a body mesh of a cube

Under «Body nets» you will find a detailed explanation. Just click on the name.

You can also open the «Representations of bodies» explanation for a more detailed overview of the representations.

Geometric bodies in everyday life and the environment

You have certainly encountered geometric bodies in your everyday life and in your environment. Think about it, can you think of one?

At the beginning you already got to know the shoe box as a geometric body. He is a square.

Objects in your everyday life are often not exact geometric bodies. A shelf or a wardrobe sometimes becomes a cuboid if you mentally omit the end panels on the edges and the handles.

Figure 11: Shelf as a cuboid

You can also discover geometric bodies in your environment. For example, imagine a tree trunk with no root or crown. That could be a cylinder.

Again, the tree trunk only resembles a cylinder and is not exactly one. Because the tree probably didn’t grow completely straight.

Have you ever taken a closer look at a dice? At first glance it appears to be a geometric cube. That’s already obvious from the name. But look at the corners. They are mostly rounded. Then a game die is not an exact geometric die, but only very similar to it. But there are also dice with corners.

But these are just examples of geometric bodies in the environment and everyday life. There are many more. Maybe you would like to keep an eye out for geometric bodies in your environment in the next few hours?

difference surface and body

In addition to the term «body», you have probably also heard of surfaces. Also in this explanation areas appeared under the properties of bodies.

One Surface is distinct from a body two-dimensional. It has no spatial content.

But a geometric body has faces. Look at Figure 1 again. The sides of a body are faces.

So remember: A surface is two-dimensional, a body is three-dimensional.

Figure 13: Difference between body and surface

Geometric solids – exercises

Here you will find basic tasks on the subject of geometric bodies. There are further tasks in the explanations for the individual bodies.

Task 1

Give the number of vertices, edges and faces of the following solids:

a) a cuboid

b) a cylinder

solution

a) Each cuboid consists of exactly six faces. They are all rectangles. The number of corners is eight. It has twelve edges.

b) A cylinder consists of three surfaces: base, top and lateral surface. It has two edges and no corners.

exercise 2

Give the name of the geometric solid in figure 14 and justify your decision.

Figure 14: geometric body

solution

The geometric body in Figure 14 is a prism. The base and the top surface (bottom and lid) are congruent and parallel. As a result, all side faces are rectangles.

In the case of a prism, the shape of the base and top surface is not important. It is important that they are identical.

Geometric Solids – The Most Important

  • A geometric body is a three-dimensional structure.
  • Geometric bodies have volume.
  • The geometric body has surfaces, corners and edges.
  • Important geometric bodies are:
  • You can draw the geometric body as an oblique image.

proof

  1. Becker et al. (2016). Formula collection up to high school – mathematics – physics – astronomy – chemistry – biology – computer science. Duden school book publisher.