21 26 27 triangle

Acute scalene triangle.

Sides: a = 21   b = 26   c = 27

Area: T = 255.1866206524
Perimeter: p = 74
Semiperimeter: s = 37

Angle ∠ A = α = 46.63876682605° = 46°38'16″ = 0.81439808666 rad
Angle ∠ B = β = 64.17548595695° = 64°10'29″ = 1.12200625965 rad
Angle ∠ C = γ = 69.187747217° = 69°11'15″ = 1.20875491905 rad

Height: ha = 24.30334482404
Height: hb = 19.63297081941
Height: hc = 18.90326819647

Median: ma = 24.33661870473
Median: mb = 20.39660780544
Median: mc = 19.39771647413

Inradius: r = 6.89769245006
Circumradius: R = 14.44223950268

Vertex coordinates: A[27; 0] B[0; 0] C[9.14881481481; 18.90326819647]
Centroid: CG[12.0499382716; 6.30108939882]
Coordinates of the circumscribed circle: U[13.5; 5.13215469509]
Coordinates of the inscribed circle: I[11; 6.89769245006]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.3622331739° = 133°21'44″ = 0.81439808666 rad
∠ B' = β' = 115.8255140431° = 115°49'31″ = 1.12200625965 rad
∠ C' = γ' = 110.813252783° = 110°48'45″ = 1.20875491905 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 = 26 ; ; c = 27 ; ;

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

p = a+b+c = 21+26+27 = 74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74 }{ 2 } = 37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37 * (37-21)(37-26)(37-27) } ; ; T = sqrt{ 65120 } = 255.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 255.19 }{ 21 } = 24.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 255.19 }{ 26 } = 19.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 255.19 }{ 27 } = 18.9 ; ;

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**2-26**2-27**2 }{ 2 * 26 * 27 } ) = 46° 38'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 64° 10'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-21**2-26**2 }{ 2 * 26 * 21 } ) = 69° 11'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 255.19 }{ 37 } = 6.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 46° 38'16" } = 14.44 ; ;




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