21 23 29 triangle

Acute scalene triangle.

Sides: a = 21   b = 23   c = 29

Area: T = 239.3376974787
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 45.86109720011° = 45°51'39″ = 0.88004249596 rad
Angle ∠ B = β = 51.81332531795° = 51°48'48″ = 0.90443118642 rad
Angle ∠ C = γ = 82.32657748194° = 82°19'33″ = 1.43768558299 rad

Height: ha = 22.79439975987
Height: hb = 20.8121910851
Height: hc = 16.50659982611

Median: ma = 23.97439441895
Median: mb = 22.55554871373
Median: mc = 16.57655844543

Inradius: r = 6.55771773914
Circumradius: R = 14.63110447983

Vertex coordinates: A[29; 0] B[0; 0] C[12.98327586207; 16.50659982611]
Centroid: CG[13.99442528736; 5.50219994204]
Coordinates of the circumscribed circle: U[14.5; 1.95438351749]
Coordinates of the inscribed circle: I[13.5; 6.55771773914]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.1399027999° = 134°8'21″ = 0.88004249596 rad
∠ B' = β' = 128.1876746821° = 128°11'12″ = 0.90443118642 rad
∠ C' = γ' = 97.67442251806° = 97°40'27″ = 1.43768558299 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 = 21 ; ; b = 23 ; ; c = 29 ; ;

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

p = a+b+c = 21+23+29 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-21)(36.5-23)(36.5-29) } ; ; T = sqrt{ 57282.19 } = 239.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 239.34 }{ 21 } = 22.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 239.34 }{ 23 } = 20.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 239.34 }{ 29 } = 16.51 ; ;

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{ 21**2-23**2-29**2 }{ 2 * 23 * 29 } ) = 45° 51'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-21**2-29**2 }{ 2 * 21 * 29 } ) = 51° 48'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-21**2-23**2 }{ 2 * 23 * 21 } ) = 82° 19'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 239.34 }{ 36.5 } = 6.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 45° 51'39" } = 14.63 ; ;




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