10 13 14 triangle

Acute scalene triangle.

Sides: a = 10   b = 13   c = 14

Area: T = 62.38553949254
Perimeter: p = 37
Semiperimeter: s = 18.5

Angle ∠ A = α = 43.27991752006° = 43°16'45″ = 0.75553641048 rad
Angle ∠ B = β = 63.02769449584° = 63°1'37″ = 1.1100027707 rad
Angle ∠ C = γ = 73.6943879841° = 73°41'38″ = 1.28662008418 rad

Height: ha = 12.47770789851
Height: hb = 9.59877530654
Height: hc = 8.91221992751

Median: ma = 12.5549900398
Median: mb = 10.28334819006
Median: mc = 9.24766210045

Inradius: r = 3.37221835095
Circumradius: R = 7.29333737222

Vertex coordinates: A[14; 0] B[0; 0] C[4.53657142857; 8.91221992751]
Centroid: CG[6.17985714286; 2.97107330917]
Coordinates of the circumscribed circle: U[7; 2.04877549297]
Coordinates of the inscribed circle: I[5.5; 3.37221835095]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.7210824799° = 136°43'15″ = 0.75553641048 rad
∠ B' = β' = 116.9733055042° = 116°58'23″ = 1.1100027707 rad
∠ C' = γ' = 106.3066120159° = 106°18'22″ = 1.28662008418 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 = 10 ; ; b = 13 ; ; c = 14 ; ;

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

p = a+b+c = 10+13+14 = 37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-10)(18.5-13)(18.5-14) } ; ; T = sqrt{ 3891.94 } = 62.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.39 }{ 10 } = 12.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.39 }{ 13 } = 9.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.39 }{ 14 } = 8.91 ; ;

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{ 10**2-13**2-14**2 }{ 2 * 13 * 14 } ) = 43° 16'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-10**2-14**2 }{ 2 * 10 * 14 } ) = 63° 1'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14**2-10**2-13**2 }{ 2 * 13 * 10 } ) = 73° 41'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.39 }{ 18.5 } = 3.37 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 43° 16'45" } = 7.29 ; ;




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