23 27 29 triangle

Acute scalene triangle.

Sides: a = 23   b = 27   c = 29

Area: T = 292.4765960551
Perimeter: p = 79
Semiperimeter: s = 39.5

Angle ∠ A = α = 48.33767776036° = 48°20'12″ = 0.8443635919 rad
Angle ∠ B = β = 61.28113004371° = 61°16'53″ = 1.07695604625 rad
Angle ∠ C = γ = 70.38219219593° = 70°22'55″ = 1.22883962721 rad

Height: ha = 25.43326922218
Height: hb = 21.66548859667
Height: hc = 20.17107559

Median: ma = 25.54989725821
Median: mb = 22.42220873248
Median: mc = 20.46333819297

Inradius: r = 7.40444546975
Circumradius: R = 15.39435728308

Vertex coordinates: A[29; 0] B[0; 0] C[11.05217241379; 20.17107559]
Centroid: CG[13.35105747126; 6.72435853]
Coordinates of the circumscribed circle: U[14.5; 5.16883734867]
Coordinates of the inscribed circle: I[12.5; 7.40444546975]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.6633222396° = 131°39'48″ = 0.8443635919 rad
∠ B' = β' = 118.7198699563° = 118°43'7″ = 1.07695604625 rad
∠ C' = γ' = 109.6188078041° = 109°37'5″ = 1.22883962721 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 = 23 ; ; b = 27 ; ; c = 29 ; ;

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

p = a+b+c = 23+27+29 = 79 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79 }{ 2 } = 39.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.5 * (39.5-23)(39.5-27)(39.5-29) } ; ; T = sqrt{ 85542.19 } = 292.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 292.48 }{ 23 } = 25.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 292.48 }{ 27 } = 21.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 292.48 }{ 29 } = 20.17 ; ;

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{ 23**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 48° 20'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-23**2-29**2 }{ 2 * 23 * 29 } ) = 61° 16'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-23**2-27**2 }{ 2 * 27 * 23 } ) = 70° 22'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 292.48 }{ 39.5 } = 7.4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 48° 20'12" } = 15.39 ; ;




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