12 13 14 triangle

Acute scalene triangle.

Sides: a = 12   b = 13   c = 14

Area: T = 72.30879352492
Perimeter: p = 39
Semiperimeter: s = 19.5

Angle ∠ A = α = 52.61768015821° = 52°37' = 0.91883364295 rad
Angle ∠ B = β = 59.40875112549° = 59°24'27″ = 1.03768566718 rad
Angle ∠ C = γ = 67.9765687163° = 67°58'32″ = 1.18663995523 rad

Height: ha = 12.05113225415
Height: hb = 11.12442977306
Height: hc = 10.33297050356

Median: ma = 12.10437184369
Median: mb = 11.30326545555
Median: mc = 10.36882206767

Inradius: r = 3.70880992435
Circumradius: R = 7.55110384596

Vertex coordinates: A[14; 0] B[0; 0] C[6.10771428571; 10.33297050356]
Centroid: CG[6.70223809524; 3.44332350119]
Coordinates of the circumscribed circle: U[7; 2.83216394223]
Coordinates of the inscribed circle: I[6.5; 3.70880992435]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.3833198418° = 127°23' = 0.91883364295 rad
∠ B' = β' = 120.5922488745° = 120°35'33″ = 1.03768566718 rad
∠ C' = γ' = 112.0244312837° = 112°1'28″ = 1.18663995523 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 = 13 ; ; c = 14 ; ;

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

p = a+b+c = 12+13+14 = 39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39 }{ 2 } = 19.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.5 * (19.5-12)(19.5-13)(19.5-14) } ; ; T = sqrt{ 5228.44 } = 72.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72.31 }{ 12 } = 12.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72.31 }{ 13 } = 11.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72.31 }{ 14 } = 10.33 ; ;

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**2-13**2-14**2 }{ 2 * 13 * 14 } ) = 52° 37' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-14**2 }{ 2 * 12 * 14 } ) = 59° 24'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 67° 58'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72.31 }{ 19.5 } = 3.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 52° 37' } = 7.55 ; ;




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