11 14 15 triangle

Acute scalene triangle.

Sides: a = 11   b = 14   c = 15

Area: T = 73.48546922835
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ B = β = 62.96443082106° = 62°57'52″ = 1.09989344895 rad
Angle ∠ C = γ = 72.62203831922° = 72°37'13″ = 1.26774647908 rad

Height: ha = 13.36108531425
Height: hb = 10.49878131834
Height: hc = 9.79879589711

Median: ma = 13.42657215821
Median: mb = 11.13655287257
Median: mc = 10.11218742081

Inradius: r = 3.67442346142
Circumradius: R = 7.85987795914

Vertex coordinates: A[15; 0] B[0; 0] C[5; 9.79879589711]
Centroid: CG[6.66766666667; 3.26659863237]
Coordinates of the circumscribed circle: U[7.5; 2.34774276702]
Coordinates of the inscribed circle: I[6; 3.67442346142]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ B' = β' = 117.0365691789° = 117°2'8″ = 1.09989344895 rad
∠ C' = γ' = 107.3879616808° = 107°22'47″ = 1.26774647908 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 = 14 ; ; c = 15 ; ;

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

p = a+b+c = 11+14+15 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-11)(20-14)(20-15) } ; ; T = sqrt{ 5400 } = 73.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 73.48 }{ 11 } = 13.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 73.48 }{ 14 } = 10.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 73.48 }{ 15 } = 9.8 ; ;

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**2-14**2-15**2 }{ 2 * 14 * 15 } ) = 44° 24'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-11**2-15**2 }{ 2 * 11 * 15 } ) = 62° 57'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-11**2-14**2 }{ 2 * 14 * 11 } ) = 72° 37'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 73.48 }{ 20 } = 3.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 44° 24'55" } = 7.86 ; ;




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