How to Recognize and Calculate the Features of a Tetrahedron
Let us first understand about Platonic Solids before we discuss Tetrahedron.
Platonic Solids
A platonic solid is a regular convex polyhedron in a three-dimensional space with identical faces consisting of congruent convex regular polygonal faces. The tetrahedron, cube, octahedron, dodecahedron, and icosahedral are the five solids that follow this criterion.
Tetrahedron
In geometry, a tetrahedron is known as a triangular pyramid. It is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. In simple words, a tetrahedron is a Platonic solid which has a three-dimensional shape having all faces as triangles.
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Tetrahedron Characteristics
A Tetrahedron will have four sides (tetrahedron faces), six edges (tetrahedron edges) and 4 corners.
All four vertices are equally distant from one another.
Three edges intersect at each vertex.
It has six symmetry planes.
A tetrahedron has no parallel faces, unlike most platonic solids.
On all of its sides, a regular tetrahedron has equilateral triangles.
Tetrahedral Structures
Right and Oblique Tetrahedrons
We can define a tetrahedron as either a right tetrahedron or an oblique tetrahedron. If the tetrahedron's apex is immediately above the base's centre, it is the right tetrahedron. If not, it is a tetrahedron that is oblique. The line segment is perpendicular to the base, which is the height of the tetrahedron, from the apex to the middle of the base of the right tetrahedron.
Regular and Irregular Tetrahedrons
It is also possible to categorise a tetrahedron as regular or irregular. If equilateral triangles are the four faces of a tetrahedron, then the tetrahedron is a regular tetrahedron. It is irregular, otherwise. All the edges of a regular tetrahedron are equal in length and are congruent to each other on all the faces of a tetrahedron. A regular tetrahedron is a proper tetrahedron as well. An irregular tetrahedron is also an oblique tetrahedron.
Regular Tetrahedron Formulas
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Area of One Face of Tetrahedron:
\[A = \frac{1}{4} \sqrt{3} a^{2}\]
Area of Tetrahedron:
\[A = \sqrt{3} a^{2}\]
Slant Height of Tetrahedron:
\[h = (\frac{\sqrt{3}}{2})a\]
Altitude of a Tetrahedron:
\[h = \frac{a\sqrt{6}}{3}\]
Volume of Tetrahedron:
\[V = \sqrt{a^{3}}{6\sqrt{2}}\]
Make Your Own Tetrahedron
Follow this procedure to make a tetrahedron on your own.
First, let’s take a sheet or paper.
We’ll make similar lines on the sheet mentioned in the paper below.
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Then we’ll cut the sheet in the edges and fold it as guided in the figure shown below to get a tetrahedral shape.
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The folded paper will form a tetrahedron.
Do You Know?
As we know, a Tetrahedron can be prepared, like all convex polyhedra, by folding a single sheet of paper. It is a polyhedron with the fewest number of faces that can be formed. As shown below, Tetrahedron also has two such different networks.
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There will be a sphere for all of the Tetrahedrons on which all four vertices will lie, and another sphere will be tangent to the faces of the Tetrahedrons.
The simplest Tetrahedron is constructed of four equal-sided triangles: one is used as the base, and the other three are fitted to it and each other to make a pyramid. But Egypt's great pyramids are not tetrahedrons. They have a square base and four triangular faces instead and are thus five-faced rather than four-faced.
Should we think of Tetrahedron as the Dice? Yes! Yes! There is an equal probability for a tetrahedron that has four equal faces to fall on any face. Actually, from all the Platonic Solids, you may make equal dice.
Solved Problems
Q1. What is the volume of a Regular tetrahedron, if it's TSA is \[25\sqrt{3}\]?
Ans: First, let’s find out the length of the edge.
So, we know that TSA of Tetrahedron is,
\[A =\sqrt{3}a^{2}\]
By substituting TSA here, we get
\[25\sqrt{3} = \sqrt{3}a^{2}\]
\[25 = a^{2}\]
Hence, a = 5 cm
Now, Volume \[V = \frac{a^{3}}{6\sqrt{2}}\]
V ≈ 14.67 cm3
This is all about different kinds of tetrahedra and their formulas. Learn how the formulas have been derived and used for calculations.
FAQs on Tetrahedron Explained: Properties, Structure, Formulas
1. What exactly is a tetrahedron and what are its main properties?
A tetrahedron is a three-dimensional shape, a type of pyramid with a triangular base. It is the simplest of all ordinary convex polyhedra. Its key properties are:
- Faces: It has 4 triangular faces.
- Vertices: It has 4 vertices (corners), where 3 faces meet at each vertex.
- Edges: It has 6 edges, where each edge connects two vertices.
- Simplicity: It is a polyhedron with the minimum possible number of faces.
2. What is the difference between a regular tetrahedron and any other tetrahedron?
The key difference lies in the shape of its faces. A regular tetrahedron is a special type where all four faces are equilateral triangles of the same size. This means all its edges are of equal length and all its face angles are 60 degrees. In a non-regular (or irregular) tetrahedron, the triangular faces can be of different shapes and sizes (e.g., scalene or isosceles triangles).
3. How do you calculate the volume and total surface area of a regular tetrahedron?
For a regular tetrahedron with an edge length 'a', you can use the following standard formulas as per the CBSE curriculum:
- Total Surface Area (TSA): The TSA is the sum of the areas of its four identical equilateral triangle faces. The formula is: TSA = √3 * a²
- Volume (V): The volume measures the space enclosed by the tetrahedron. The formula is: V = a³ / (6√2)
These formulas are important for solving problems in solid geometry.
4. What does the 'net' of a tetrahedron look like?
The net of a tetrahedron is a two-dimensional shape that can be folded to form the 3D tetrahedron. The most common net consists of four triangles joined at their edges. Typically, it looks like one central triangle with three other triangles attached to each of its sides. When you fold up the three outer triangles, they meet at a single point, the apex, forming the tetrahedron.
5. How is a tetrahedron different from a triangular prism?
A tetrahedron and a triangular prism are both polyhedra but differ significantly in their structure. A tetrahedron is a pyramid with a triangular base and three triangular faces that meet at a single apex. In contrast, a triangular prism has two identical triangular bases and three rectangular faces connecting them. A tetrahedron has 4 faces, 4 vertices, and 6 edges, while a prism has 5 faces, 6 vertices, and 9 edges.
6. Why is the tetrahedron shape important in real-world examples, such as chemistry?
The tetrahedron shape is fundamental in science and engineering for its stability and efficiency. A prime example is in chemistry, where it describes the tetrahedral molecular geometry. For instance, in a methane molecule (CH₄), the central carbon atom is at the center of a tetrahedron, and the four hydrogen atoms are at the four vertices. This arrangement minimises the repulsion between the electron pairs, making it a very stable configuration.
7. Is it possible to perfectly fill a cube with tetrahedrons?
Yes, a cube can be perfectly divided (or 'tessellated') into smaller tetrahedrons. A common way to do this is to divide the cube into five tetrahedrons. This is achieved by selecting four of the cube's eight vertices that do not share an edge and using them to form a large, regular tetrahedron in the center. The four corner pieces left over are also smaller, irregular tetrahedrons.
8. Can any four points in space form a tetrahedron?
No, not any four points can form a tetrahedron. The crucial condition is that the four points must be non-coplanar, meaning they cannot all lie on the same flat plane. If the four points were on the same plane, you would only be able to form a flat quadrilateral or a triangle. A tetrahedron requires one point to be elevated out of the plane formed by the other three, creating the necessary height and volume for the 3D shape.
