11 13 14 triangle

Acute scalene triangle.

Sides: a = 11   b = 13   c = 14

Area: T = 67.52877720645
Perimeter: p = 38
Semiperimeter: s = 19

Angle ∠ A = α = 47.90774859842° = 47°54'27″ = 0.83661433668 rad
Angle ∠ B = β = 61.28106643991° = 61°16'50″ = 1.07695493616 rad
Angle ∠ C = γ = 70.81218496167° = 70°48'43″ = 1.23658999252 rad

Height: ha = 12.2787776739
Height: hb = 10.38988880099
Height: hc = 9.64768245806

Median: ma = 12.33989626793
Median: mb = 10.78219293264
Median: mc = 9.79879589711

Inradius: r = 3.55440932666
Circumradius: R = 7.4121765333

Vertex coordinates: A[14; 0] B[0; 0] C[5.28657142857; 9.64768245806]
Centroid: CG[6.42985714286; 3.21656081935]
Coordinates of the circumscribed circle: U[7; 2.43660347598]
Coordinates of the inscribed circle: I[6; 3.55440932666]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.0932514016° = 132°5'33″ = 0.83661433668 rad
∠ B' = β' = 118.7199335601° = 118°43'10″ = 1.07695493616 rad
∠ C' = γ' = 109.1888150383° = 109°11'17″ = 1.23658999252 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 = 11 ; ; b = 13 ; ; c = 14 ; ;

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

p = a+b+c = 11+13+14 = 38 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-11)(19-13)(19-14) } ; ; T = sqrt{ 4560 } = 67.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 67.53 }{ 11 } = 12.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 67.53 }{ 13 } = 10.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 67.53 }{ 14 } = 9.65 ; ;

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{ 11**2-13**2-14**2 }{ 2 * 13 * 14 } ) = 47° 54'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-11**2-14**2 }{ 2 * 11 * 14 } ) = 61° 16'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14**2-11**2-13**2 }{ 2 * 13 * 11 } ) = 70° 48'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 67.53 }{ 19 } = 3.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 47° 54'27" } = 7.41 ; ;




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