9 14 20 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 14   c = 20

Area: T = 54.98657936198
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 23.12660741872° = 23°7'34″ = 0.40436261376 rad
Angle ∠ B = β = 37.65884620062° = 37°39'30″ = 0.65772641532 rad
Angle ∠ C = γ = 119.2155463807° = 119°12'56″ = 2.08107023627 rad

Height: ha = 12.21990652488
Height: hb = 7.85551133743
Height: hc = 5.4998579362

Median: ma = 16.66658333125
Median: mb = 13.83883525031
Median: mc = 6.2054836823

Inradius: r = 2.5577478773
Circumradius: R = 11.45875049031

Vertex coordinates: A[20; 0] B[0; 0] C[7.125; 5.4998579362]
Centroid: CG[9.04216666667; 1.83328597873]
Coordinates of the circumscribed circle: U[10; -5.59223535837]
Coordinates of the inscribed circle: I[7.5; 2.5577478773]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.8743925813° = 156°52'26″ = 0.40436261376 rad
∠ B' = β' = 142.3421537994° = 142°20'30″ = 0.65772641532 rad
∠ C' = γ' = 60.78545361934° = 60°47'4″ = 2.08107023627 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 = 9 ; ; b = 14 ; ; c = 20 ; ;

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

p = a+b+c = 9+14+20 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-9)(21.5-14)(21.5-20) } ; ; T = sqrt{ 3023.44 } = 54.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 54.99 }{ 9 } = 12.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 54.99 }{ 14 } = 7.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 54.99 }{ 20 } = 5.5 ; ;

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{ 9**2-14**2-20**2 }{ 2 * 14 * 20 } ) = 23° 7'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-9**2-20**2 }{ 2 * 9 * 20 } ) = 37° 39'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-9**2-14**2 }{ 2 * 14 * 9 } ) = 119° 12'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 54.99 }{ 21.5 } = 2.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 23° 7'34" } = 11.46 ; ;




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