Formula For The Volume Of A Octagon
In geometry, an octagon is a polygon with eight sides. A regular octagon has eight equal sides and equal angles. The regular octagon is commonly recognized from stop signs. An octahedron is an eight-sided polyhedron. A regular octahedron has eight triangles with edges of equal length. It is effectively two square pyramids meeting at their bases.
Octagon Area Formula
The formula for the area of a regular octagon with sides of length "a" is 2(1+sqrt(2))a^2, where "sqrt" indicates the square root.
Derivation
An octagon can be viewed as 4 rectangles, one square in the center and four isosceles triangles in the corners.
The square is of area a^2.
The triangles have sides a, a/sqrt(2) and a/sqrt(2), by the Pythagorean theorem. Therefore, each has an area of a^2/4.
The rectangles are of area a * a/sqrt(2).
The sum of these 9 areas is 2a^2 (1 + sqrt(2)).
Octahedron Volume Formula
The formula for the volume of a regular octahedron of sides "a" is a^3 * sqrt(2)/3.
Derivation
The area of a four-sided pyramid is area of base height / 3. The area of a regular octagon is therefore 2 base * height / 3.
Base = a^2 trivially.
Pick two adjacent vertices, say "F" and "C." "O" is at the center. FOC is an isosceles right triangle with base "a," so OC and OF have length a/sqrt(2) by the Pythagorean theorem. So height = a/sqrt(2).
So the volume of a regular octahedron is 2 (a^2) a/sqrt(2) / 3 = a^3 * sqrt(2) / 3.
Surface Area
The regular octahedron's surface is the area of an equilateral triangle of side "a" times 8 faces.
To use the Pythagorean theorem, drop a line from the apex to the base. This creates two right triangles, with the hypotenuse of length "a" and one side length "a/2." Therefore, the third side must be sqrt[a^2 - a^2/4] = sqrt(3)a/2. So the area of an equilateral triangle is height base/2 = sqrt(3)a/2 a/2 = sqrt(3)a^2/4.
With 8 sides, the surface area of a regular octahedron is 2 sqrt(3) a^2.
References
Cite This Article
MLA
Dohrman, Paul. "Formula For The Volume Of A Octagon" sciencing.com, https://www.sciencing.com/formula-volume-octagon-5250324/. 24 April 2017.
APA
Dohrman, Paul. (2017, April 24). Formula For The Volume Of A Octagon. sciencing.com. Retrieved from https://www.sciencing.com/formula-volume-octagon-5250324/
Chicago
Dohrman, Paul. Formula For The Volume Of A Octagon last modified March 24, 2022. https://www.sciencing.com/formula-volume-octagon-5250324/