13 26 27 triangle

Acute scalene triangle.

Sides: a = 13   b = 26   c = 27

Area: T = 166.4933243106
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 28.31663682245° = 28°18'59″ = 0.49442138577 rad
Angle ∠ B = β = 71.56443535993° = 71°33'52″ = 1.24990335974 rad
Angle ∠ C = γ = 80.11992781761° = 80°7'9″ = 1.39883451985 rad

Height: ha = 25.61443450933
Height: hb = 12.80771725466
Height: hc = 12.33328328227

Median: ma = 25.69553303151
Median: mb = 16.73332005307
Median: mc = 15.5

Inradius: r = 5.04552497911
Circumradius: R = 13.70332588076

Vertex coordinates: A[27; 0] B[0; 0] C[4.11111111111; 12.33328328227]
Centroid: CG[10.37703703704; 4.11109442742]
Coordinates of the circumscribed circle: U[13.5; 2.35114467776]
Coordinates of the inscribed circle: I[7; 5.04552497911]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.6843631775° = 151°41'1″ = 0.49442138577 rad
∠ B' = β' = 108.4365646401° = 108°26'8″ = 1.24990335974 rad
∠ C' = γ' = 99.88107218239° = 99°52'51″ = 1.39883451985 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 = 13 ; ; b = 26 ; ; c = 27 ; ;

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

p = a+b+c = 13+26+27 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-13)(33-26)(33-27) } ; ; T = sqrt{ 27720 } = 166.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 166.49 }{ 13 } = 25.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 166.49 }{ 26 } = 12.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 166.49 }{ 27 } = 12.33 ; ;

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{ 13**2-26**2-27**2 }{ 2 * 26 * 27 } ) = 28° 18'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 71° 33'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-26**2 }{ 2 * 26 * 13 } ) = 80° 7'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 166.49 }{ 33 } = 5.05 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 28° 18'59" } = 13.7 ; ;




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