11 13 21 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 13   c = 21

Area: T = 60.72222158687
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 26.41438316656° = 26°24'50″ = 0.46110083306 rad
Angle ∠ B = β = 31.7187633276° = 31°43'3″ = 0.55435771316 rad
Angle ∠ C = γ = 121.8698535058° = 121°52'7″ = 2.12770071914 rad

Height: ha = 11.04404028852
Height: hb = 9.34218793644
Height: hc = 5.7833068178

Median: ma = 16.57655844543
Median: mb = 15.45215371404
Median: mc = 5.89549130613

Inradius: r = 2.69987651497
Circumradius: R = 12.36436792442

Vertex coordinates: A[21; 0] B[0; 0] C[9.35771428571; 5.7833068178]
Centroid: CG[10.1199047619; 1.92876893927]
Coordinates of the circumscribed circle: U[10.5; -6.52876768038]
Coordinates of the inscribed circle: I[9.5; 2.69987651497]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.5866168334° = 153°35'10″ = 0.46110083306 rad
∠ B' = β' = 148.2822366724° = 148°16'57″ = 0.55435771316 rad
∠ C' = γ' = 58.13114649415° = 58°7'53″ = 2.12770071914 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 = 11 ; ; b = 13 ; ; c = 21 ; ;

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

p = a+b+c = 11+13+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-11)(22.5-13)(22.5-21) } ; ; T = sqrt{ 3687.19 } = 60.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60.72 }{ 11 } = 11.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60.72 }{ 13 } = 9.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60.72 }{ 21 } = 5.78 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13**2+21**2-11**2 }{ 2 * 13 * 21 } ) = 26° 24'50" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11**2+21**2-13**2 }{ 2 * 11 * 21 } ) = 31° 43'3" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 11**2+13**2-21**2 }{ 2 * 11 * 13 } ) = 121° 52'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60.72 }{ 22.5 } = 2.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 26° 24'50" } = 12.36 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 21**2 - 11**2 } }{ 2 } = 16.576 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 11**2 - 13**2 } }{ 2 } = 15.452 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 11**2 - 21**2 } }{ 2 } = 5.895 ; ;
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