# Facts about Regular Polygons and Polyhedra

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## Facts about Regular Polygons and Polyhedra

The following legend pertains to the table below it:

s = length of a side A = area d = diagonal

R = radius of circumscribed circle or sphere V = volume a = length of an edge

r = radius of inscribed circle h = height S = surface area

Triangle A = ((s^2)*sqrt(3)) / 4 Tetrahedron V = (1/12) * sqrt(2) * a^3

R = (1/3) * s * sqrt(3) S = a^3 * sqrt(3)

r = (1/6) * s * sqrt(3) R = (1/4) * a * sqrt(6)

Isosceles Area = (c/4)(sqrt(4a^2 - c^2)

where (c is the base, a are the legs) r = (1/12) * a * sqrt(6)

Square A = s^2 Cube V = a^3

R = (1/2) * s * sqrt(2) S = 6 * a^2

r = (1/2) * s R = (1/2) * a * sqrt(3)

r = (1/2) * a

Pentagon A = (s^2 / 4) ( sqrt( 25 + 10 * sqrt(5) ) ) Octahedron V = (1/3) * a^3 * sqrt(2)

R = (1/10) * s * sqrt (50 + 10 * sqrt(5) ) S = 2 * a^2 * sqrt(3)

r = (1/10) * s * sqrt (25 + 10 * sqrt(5) ) R = (1/2) * a * sqrt(2)

r = (1/6) * a * sqrt(6)

Hexagon A = (3/2) * s^2 * sqrt(3) Dodecahedron V = (1/4) * a^3 * (15 + 7 * sqrt(5) )

R = s S = 3 * a^2 * sqrt(25 + 10 * sqrt(5) )

r = (1/2) * s * sqrt(3) R = (1/4) * a * ( sqrt(3) + sqrt(15) )

r = (1/4) * a * sqrt( (50 + 22 * sqrt(5)) / 5 )

Octagon A = 2 * s^2 * (1 + sqrt(2) ) Icosahedron V = (5/12) * a^3 * (3 + sqrt (5) )

R = (1/2) * a * sqrt( 4 + 2 * sqrt(2) ) S = 5 * a^2 * sqrt(3)

r = (1/2) * a * (1 + sqrt(2) ) R = (1/4) * a * sqrt(10 + 2 * sqrt(5) )

r = (1/2) * a * sqrt( (7 + 3 * sqrt (5)) / 6 )

Decagon A = (5/2) * s^2 * sqrt (5 + 2 * sqrt(5)) Sphere V = (4/3) * pi * a^3

R = (1/2) * a * (1 + sqrt(5) ) S = 4 * pi * a^2

r = (1/2) * a * sqrt( 5 + 2 * sqrt(5) )

Cone S = (pi) * r * sqrt(r^2 + h^2) Radius of circle inscribed in a triangle R = sqrt[s * (s - a) * (s - b) * (s - c)] / s

where s = (1/2) * (a + b + c)

s = length of a side A = area d = diagonal

R = radius of circumscribed circle or sphere V = volume a = length of an edge

r = radius of inscribed circle h = height S = surface area

Triangle A = ((s^2)*sqrt(3)) / 4 Tetrahedron V = (1/12) * sqrt(2) * a^3

R = (1/3) * s * sqrt(3) S = a^3 * sqrt(3)

r = (1/6) * s * sqrt(3) R = (1/4) * a * sqrt(6)

Isosceles Area = (c/4)(sqrt(4a^2 - c^2)

where (c is the base, a are the legs) r = (1/12) * a * sqrt(6)

Square A = s^2 Cube V = a^3

R = (1/2) * s * sqrt(2) S = 6 * a^2

r = (1/2) * s R = (1/2) * a * sqrt(3)

r = (1/2) * a

Pentagon A = (s^2 / 4) ( sqrt( 25 + 10 * sqrt(5) ) ) Octahedron V = (1/3) * a^3 * sqrt(2)

R = (1/10) * s * sqrt (50 + 10 * sqrt(5) ) S = 2 * a^2 * sqrt(3)

r = (1/10) * s * sqrt (25 + 10 * sqrt(5) ) R = (1/2) * a * sqrt(2)

r = (1/6) * a * sqrt(6)

Hexagon A = (3/2) * s^2 * sqrt(3) Dodecahedron V = (1/4) * a^3 * (15 + 7 * sqrt(5) )

R = s S = 3 * a^2 * sqrt(25 + 10 * sqrt(5) )

r = (1/2) * s * sqrt(3) R = (1/4) * a * ( sqrt(3) + sqrt(15) )

r = (1/4) * a * sqrt( (50 + 22 * sqrt(5)) / 5 )

Octagon A = 2 * s^2 * (1 + sqrt(2) ) Icosahedron V = (5/12) * a^3 * (3 + sqrt (5) )

R = (1/2) * a * sqrt( 4 + 2 * sqrt(2) ) S = 5 * a^2 * sqrt(3)

r = (1/2) * a * (1 + sqrt(2) ) R = (1/4) * a * sqrt(10 + 2 * sqrt(5) )

r = (1/2) * a * sqrt( (7 + 3 * sqrt (5)) / 6 )

Decagon A = (5/2) * s^2 * sqrt (5 + 2 * sqrt(5)) Sphere V = (4/3) * pi * a^3

R = (1/2) * a * (1 + sqrt(5) ) S = 4 * pi * a^2

r = (1/2) * a * sqrt( 5 + 2 * sqrt(5) )

Cone S = (pi) * r * sqrt(r^2 + h^2) Radius of circle inscribed in a triangle R = sqrt[s * (s - a) * (s - b) * (s - c)] / s

where s = (1/2) * (a + b + c)

**HighTec**- Posts : 49

Join date : 2007-11-17

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