Regular Tetrahedron Formula, Definition, Area, Volume

Regular Tetrahedron Formula: A regular tetrahedron consists of four equilateral triangular faces, all of which are identical and congruent to one another. This three-dimensional figure, also termed a triangular pyramid, comprises 4 faces, 6 edges, and 4 vertices. It's depicted in the figure below.

Formula for the Area of One Face of a Regular Tetrahedron

Here, 'x' represents the side length of a regular tetrahedron.

Height on the slant of a Regular Tetrahedron

Here, 'a' stands for the base of the triangular face.

The height of a Regular Tetrahedron

Here, 'a' represents the base of the triangular face.

The formula for the Total Surface Area of a Regular Tetrahedron:

As a regular tetrahedron consists of four identical equilateral triangles, its total surface area is the combined sum of these individual areas. The area of an equilateral triangle with a side length 'x' is calculated as follows:

Area of equilateral triangle =√3x²

Total surface area of  regular tetrahedron TSA = 4*√3x²

where x is the length of the side of a regular tetrahedron.

Volume

V =(a³√2) / 12

where x is the length of the side of a regular tetrahedron.

Regular Tetrahedron Formula Solved Examples

Example 1: Find the Total Surface Area (TSA) of a tetrahedron with a side length of 4 cm.

Solution:

TSA of a tetrahedron = √3x²

Here, x = 4 cm

⇒ TSA = √3 x (4)²

= 27.712 cm²

Example 2: Determine the volume of a tetrahedron with a side length of 10 cm.

Solution:

Volume of a tetrahedron = (a³√2) / 12

Here, a = 10 cm

⇒ V = (10³√2) / 12

= 117.85 cm³

Example 3: Compute the Total Surface Area (TSA) of a tetrahedron with a side length of 10 cm.

Solution:

TSA of a tetrahedron = √3x²

Here, x = 10 cm

⇒ TSA = √3 x (10)²

= 173.20 cm²

Example 4: Calculate the Total Surface Area (TSA) of a tetrahedron with a side length of 30 cm.

Solution:

TSA of a tetrahedron = √3x²

Here, x = 30 cm

⇒ TSA = √3 x (30)²

= 1558.84 cm²

Example 5: Determine the volume of a tetrahedron with a side length of 20 cm.

Solution:

Volume of a tetrahedron = (a³√2) / 12

Here, a = 20 cm

⇒ V = (20³√2) / 12

= 942.809 cm³

Example 6: Calculate the volume of a tetrahedron with a side length of 50 cm.

Solution:

Volume of a tetrahedron = (a³√2) / 12

Here, a = 50 cm

⇒ V = (50³√2) / 12

= 14731.39 cm³

Example 7: Calculate the Total Surface Area (TSA) of a regular tetrahedron with a side length of 15 cm.

Solution: The formula for the Total Surface Area (TSA) of a regular tetrahedron is given as: √3x²

Given that x = 15 cm, Substitute the value into the formula:

√3×( 15 ) 2

TSA= √3 ​ ×(15) 2

TSA= √3​ ×225

TSA≈389.71cm 2

Example 8: Find the volume of a regular tetrahedron with a side length of 12 cm.

Solution: The formula for the volume of a regular tetrahedron is (a³√2) / 12 ​ .

Given that a = 12 cm, Substitute the value into the formula: (12³√2) / 12

V= 12³√2) / 12  ​ ​

V= 1728 √2 / 12 ​ ​

V≈144 cm 3

The regular tetrahedron, a three-dimensional figure, is composed of four identical equilateral triangular faces, forming a triangular pyramid. It comprises 4 faces, 6 edges, and 4 vertices.

Formulas related to the regular tetrahedron:

Area of One Face: The area of a single face is given by the formula Area of an equilateral triangle = √3x² , where 'x' represents the side length of the tetrahedron.

Slant Height: Denoted by 'a', it signifies the base of the triangular face. Altitude: 'a' is the height of the tetrahedron, which coincides with the base of the triangular face.

Total Surface Area (TSA): The total surface area of the tetrahedron is calculated as TSA=4*√3x², with 'x' representing the side length.

Volume: The volume formula for the tetrahedron is V= (a³√2) / 12  ​ ​ , where 'a' signifies the side length.

Understanding these formulas allows for the precise calculation of the surface area and volume of a regular tetrahedron based on the given side lengths. These calculations prove valuable in geometry and engineering when dealing with triangular-based pyramids.

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