5 17 21 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 17   c = 21

Area: T = 28.25222123028
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 9.10768257876° = 9°6'25″ = 0.15989440944 rad
Angle ∠ B = β = 32.55769318375° = 32°33'25″ = 0.56882256549 rad
Angle ∠ C = γ = 138.3366242375° = 138°20'10″ = 2.41444229043 rad

Height: ha = 11.30108849211
Height: hb = 3.32437896827
Height: hc = 2.6910686886

Median: ma = 18.94106969249
Median: mb = 12.67987223331
Median: mc = 6.83773971656

Inradius: r = 1.31440563862
Circumradius: R = 15.79552232278

Vertex coordinates: A[21; 0] B[0; 0] C[4.21442857143; 2.6910686886]
Centroid: CG[8.40547619048; 0.89768956287]
Coordinates of the circumscribed circle: U[10.5; -11.87999608819]
Coordinates of the inscribed circle: I[4.5; 1.31440563862]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.8933174212° = 170°53'35″ = 0.15989440944 rad
∠ B' = β' = 147.4433068163° = 147°26'35″ = 0.56882256549 rad
∠ C' = γ' = 41.6643757625° = 41°39'50″ = 2.41444229043 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 = 5 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 5+17+21 = 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 * (21.5-5)(21.5-17)(21.5-21) } ; ; T = sqrt{ 798.19 } = 28.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.25 }{ 5 } = 11.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.25 }{ 17 } = 3.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.25 }{ 21 } = 2.69 ; ;

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{ 5**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 9° 6'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-5**2-21**2 }{ 2 * 5 * 21 } ) = 32° 33'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-5**2-17**2 }{ 2 * 17 * 5 } ) = 138° 20'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.25 }{ 21.5 } = 1.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 9° 6'25" } = 15.8 ; ;




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