13 19 22 triangle

Acute scalene triangle.

Sides: a = 13   b = 19   c = 22

Area: T = 122.9633409192
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 36.03994162329° = 36°2'22″ = 0.62990064738 rad
Angle ∠ B = β = 59.30435587083° = 59°18'13″ = 1.03550423576 rad
Angle ∠ C = γ = 84.65770250587° = 84°39'25″ = 1.47875438222 rad

Height: ha = 18.91774475679
Height: hb = 12.9443516757
Height: hc = 11.17884917447

Median: ma = 19.5
Median: mb = 15.37704261489
Median: mc = 12

Inradius: r = 4.55442003404
Circumradius: R = 11.04880020758

Vertex coordinates: A[22; 0] B[0; 0] C[6.63663636364; 11.17884917447]
Centroid: CG[9.54554545455; 3.72661639149]
Coordinates of the circumscribed circle: U[11; 1.02987613269]
Coordinates of the inscribed circle: I[8; 4.55442003404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.9610583767° = 143°57'38″ = 0.62990064738 rad
∠ B' = β' = 120.6966441292° = 120°41'47″ = 1.03550423576 rad
∠ C' = γ' = 95.34329749413° = 95°20'35″ = 1.47875438222 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 = 19 ; ; c = 22 ; ;

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

p = a+b+c = 13+19+22 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-13)(27-19)(27-22) } ; ; T = sqrt{ 15120 } = 122.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 122.96 }{ 13 } = 18.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 122.96 }{ 19 } = 12.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 122.96 }{ 22 } = 11.18 ; ;

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**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 36° 2'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-22**2 }{ 2 * 13 * 22 } ) = 59° 18'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 84° 39'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 122.96 }{ 27 } = 4.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 36° 2'22" } = 11.05 ; ;




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