9 21 28 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 21   c = 28

Area: T = 68.11875454637
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 13.39767268735° = 13°23'48″ = 0.23438169929 rad
Angle ∠ B = β = 32.72655443586° = 32°43'32″ = 0.57111684986 rad
Angle ∠ C = γ = 133.8787728768° = 133°52'40″ = 2.33766071621 rad

Height: ha = 15.13772323253
Height: hb = 6.48773852823
Height: hc = 4.86655389617

Median: ma = 24.33661870473
Median: mb = 17.9511323071
Median: mc = 8.06222577483

Inradius: r = 2.34988808781
Circumradius: R = 19.42223087604

Vertex coordinates: A[28; 0] B[0; 0] C[7.57114285714; 4.86655389617]
Centroid: CG[11.85771428571; 1.62218463206]
Coordinates of the circumscribed circle: U[14; -13.46220235324]
Coordinates of the inscribed circle: I[8; 2.34988808781]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.6033273127° = 166°36'12″ = 0.23438169929 rad
∠ B' = β' = 147.2744455641° = 147°16'28″ = 0.57111684986 rad
∠ C' = γ' = 46.12222712321° = 46°7'20″ = 2.33766071621 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 = 21 ; ; c = 28 ; ;

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

p = a+b+c = 9+21+28 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-9)(29-21)(29-28) } ; ; T = sqrt{ 4640 } = 68.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68.12 }{ 9 } = 15.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68.12 }{ 21 } = 6.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68.12 }{ 28 } = 4.87 ; ;

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**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 13° 23'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-9**2-28**2 }{ 2 * 9 * 28 } ) = 32° 43'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-9**2-21**2 }{ 2 * 21 * 9 } ) = 133° 52'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68.12 }{ 29 } = 2.35 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 13° 23'48" } = 19.42 ; ;




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