12 12 12 triangle

Equilateral triangle.

Sides: a = 12   b = 12   c = 12

Area: T = 62.35438290725
Perimeter: p = 36
Semiperimeter: s = 18

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 10.39223048454
Height: hb = 10.39223048454
Height: hc = 10.39223048454

Median: ma = 10.39223048454
Median: mb = 10.39223048454
Median: mc = 10.39223048454

Inradius: r = 3.46441016151
Circumradius: R = 6.92882032303

Vertex coordinates: A[12; 0] B[0; 0] C[6; 10.39223048454]
Centroid: CG[6; 3.46441016151]
Coordinates of the circumscribed circle: U[6; 3.46441016151]
Coordinates of the inscribed circle: I[6; 3.46441016151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 12 ; ; b = 12 ; ; c = 12 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 12+12+12 = 36 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36 }{ 2 } = 18 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18 * (18-12)(18-12)(18-12) } ; ; T = sqrt{ 3888 } = 62.35 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.35 }{ 12 } = 10.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.35 }{ 12 } = 10.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.35 }{ 12 } = 10.39 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 60° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.35 }{ 18 } = 3.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 60° } = 6.93 ; ;




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