45 45 45 triangle

Equilateral triangle.

Sides: a = 45   b = 45   c = 45

Area: T = 876.8510721332
Perimeter: p = 135
Semiperimeter: s = 67.5

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

Height: ha = 38.97111431703
Height: hb = 38.97111431703
Height: hc = 38.97111431703

Median: ma = 38.97111431703
Median: mb = 38.97111431703
Median: mc = 38.97111431703

Inradius: r = 12.99903810568
Circumradius: R = 25.98107621135

Vertex coordinates: A[45; 0] B[0; 0] C[22.5; 38.97111431703]
Centroid: CG[22.5; 12.99903810568]
Coordinates of the circumscribed circle: U[22.5; 12.99903810568]
Coordinates of the inscribed circle: I[22.5; 12.99903810568]

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?

a = 45 ; ; b = 45 ; ; c = 45 ; ;

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

p = a+b+c = 45+45+45 = 135 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 135 }{ 2 } = 67.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 67.5 * (67.5-45)(67.5-45)(67.5-45) } ; ; T = sqrt{ 768867.19 } = 876.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 876.85 }{ 45 } = 38.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 876.85 }{ 45 } = 38.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 876.85 }{ 45 } = 38.97 ; ;

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{ 45**2-45**2-45**2 }{ 2 * 45 * 45 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-45**2-45**2 }{ 2 * 45 * 45 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45**2-45**2-45**2 }{ 2 * 45 * 45 } ) = 60° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 876.85 }{ 67.5 } = 12.99 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 60° } = 25.98 ; ;




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