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: h_a = 11.04404028852
Height: h_b = 9.34218793644
Height: h_c = 5.7833068178

Median: m_a = 16.57655844543
Median: m_b = 15.45215371404
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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