9 21 29 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 21   c = 29

Area: T = 50.69770166775
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 9.58439574325° = 9°35'2″ = 0.16772716126 rad
Angle ∠ B = β = 22.8660133051° = 22°51'36″ = 0.39989845892 rad
Angle ∠ C = γ = 147.5565909516° = 147°33'21″ = 2.57553364518 rad

Height: ha = 11.26660037061
Height: hb = 4.82882873026
Height: hc = 3.49663459778

Median: ma = 24.91548550066
Median: mb = 18.72883208003
Median: mc = 7.12439034244

Inradius: r = 1.71985429382
Circumradius: R = 27.0288217631

Vertex coordinates: A[29; 0] B[0; 0] C[8.29331034483; 3.49663459778]
Centroid: CG[12.43110344828; 1.16554486593]
Coordinates of the circumscribed circle: U[14.5; -22.81095275774]
Coordinates of the inscribed circle: I[8.5; 1.71985429382]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.4166042567° = 170°24'58″ = 0.16772716126 rad
∠ B' = β' = 157.1439866949° = 157°8'24″ = 0.39989845892 rad
∠ C' = γ' = 32.44440904835° = 32°26'39″ = 2.57553364518 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 = 29 ; ;

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

p = a+b+c = 9+21+29 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-9)(29.5-21)(29.5-29) } ; ; T = sqrt{ 2570.19 } = 50.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50.7 }{ 9 } = 11.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50.7 }{ 21 } = 4.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50.7 }{ 29 } = 3.5 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 21**2+29**2-9**2 }{ 2 * 21 * 29 } ) = 9° 35'2" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+29**2-21**2 }{ 2 * 9 * 29 } ) = 22° 51'36" ; ; gamma = 180° - alpha - beta = 180° - 9° 35'2" - 22° 51'36" = 147° 33'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50.7 }{ 29.5 } = 1.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9 }{ 2 * sin 9° 35'2" } = 27.03 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 29**2 - 9**2 } }{ 2 } = 24.915 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29**2+2 * 9**2 - 21**2 } }{ 2 } = 18.728 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 9**2 - 29**2 } }{ 2 } = 7.124 ; ;
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