13 14 21 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 14   c = 21

Area: T = 88.99443818451
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 37.25879160013° = 37°15'29″ = 0.65502733067 rad
Angle ∠ B = β = 40.69105605975° = 40°41'26″ = 0.71101842569 rad
Angle ∠ C = γ = 102.0521523401° = 102°3'5″ = 1.781113509 rad

Height: ha = 13.69114433608
Height: hb = 12.71334831207
Height: hc = 8.47656554138

Median: ma = 16.62107701386
Median: mb = 16
Median: mc = 8.5

Inradius: r = 3.70880992435
Circumradius: R = 10.73766328097

Vertex coordinates: A[21; 0] B[0; 0] C[9.85771428571; 8.47656554138]
Centroid: CG[10.28657142857; 2.82552184713]
Coordinates of the circumscribed circle: U[10.5; -2.24217145427]
Coordinates of the inscribed circle: I[10; 3.70880992435]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.7422083999° = 142°44'31″ = 0.65502733067 rad
∠ B' = β' = 139.3099439402° = 139°18'34″ = 0.71101842569 rad
∠ C' = γ' = 77.94884765988° = 77°56'55″ = 1.781113509 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 = 13 ; ; b = 14 ; ; c = 21 ; ;

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

p = a+b+c = 13+14+21 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-13)(24-14)(24-21) } ; ; T = sqrt{ 7920 } = 88.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 88.99 }{ 13 } = 13.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 88.99 }{ 14 } = 12.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 88.99 }{ 21 } = 8.48 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 14**2+21**2-13**2 }{ 2 * 14 * 21 } ) = 37° 15'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+21**2-14**2 }{ 2 * 13 * 21 } ) = 40° 41'26" ; ; gamma = 180° - alpha - beta = 180° - 37° 15'29" - 40° 41'26" = 102° 3'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 88.99 }{ 24 } = 3.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13 }{ 2 * sin 37° 15'29" } = 10.74 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 21**2 - 13**2 } }{ 2 } = 16.621 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 13**2 - 14**2 } }{ 2 } = 16 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 13**2 - 21**2 } }{ 2 } = 8.5 ; ;
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