14 19 21 triangle

Acute scalene triangle.

Sides: a = 14   b = 19   c = 21

Area: T = 129.8799845917
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 40.58988006955° = 40°35'20″ = 0.70884082116 rad
Angle ∠ B = β = 62.00554382677° = 62°20″ = 1.08221990519 rad
Angle ∠ C = γ = 77.40657610367° = 77°24'21″ = 1.35109853901 rad

Height: ha = 18.5432835131
Height: hb = 13.66331416754
Height: hc = 12.36218900873

Median: ma = 18.76216630393
Median: mb = 15.10879449297
Median: mc = 12.97111217711

Inradius: r = 4.80774017006
Circumradius: R = 10.7598872556

Vertex coordinates: A[21; 0] B[0; 0] C[6.57114285714; 12.36218900873]
Centroid: CG[9.19904761905; 4.12106300291]
Coordinates of the circumscribed circle: U[10.5; 2.34659195799]
Coordinates of the inscribed circle: I[8; 4.80774017006]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.4111199304° = 139°24'40″ = 0.70884082116 rad
∠ B' = β' = 117.9954561732° = 117°59'40″ = 1.08221990519 rad
∠ C' = γ' = 102.5944238963° = 102°35'39″ = 1.35109853901 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 = 14 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 14+19+21 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-14)(27-19)(27-21) } ; ; T = sqrt{ 16848 } = 129.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.8 }{ 14 } = 18.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.8 }{ 19 } = 13.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.8 }{ 21 } = 12.36 ; ;

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{ 14**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 40° 35'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-14**2-21**2 }{ 2 * 14 * 21 } ) = 62° 20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-14**2-19**2 }{ 2 * 19 * 14 } ) = 77° 24'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.8 }{ 27 } = 4.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 40° 35'20" } = 10.76 ; ;




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