13 21 30 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 21   c = 30

Area: T = 115.6554658358
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 21.54404283647° = 21°32'26″ = 0.37659513973 rad
Angle ∠ B = β = 36.37773613942° = 36°22'38″ = 0.63549047295 rad
Angle ∠ C = γ = 122.0822210241° = 122°4'56″ = 2.13107365268 rad

Height: ha = 17.79330243628
Height: hb = 11.01547293675
Height: hc = 7.71103105572

Median: ma = 25.0654915719
Median: mb = 20.5977329924
Median: mc = 8.944427191

Inradius: r = 3.61442080737
Circumradius: R = 17.70435670596

Vertex coordinates: A[30; 0] B[0; 0] C[10.46766666667; 7.71103105572]
Centroid: CG[13.48988888889; 2.57701035191]
Coordinates of the circumscribed circle: U[15; -9.40329934932]
Coordinates of the inscribed circle: I[11; 3.61442080737]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.4659571635° = 158°27'34″ = 0.37659513973 rad
∠ B' = β' = 143.6232638606° = 143°37'22″ = 0.63549047295 rad
∠ C' = γ' = 57.91877897589° = 57°55'4″ = 2.13107365268 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 = 21 ; ; c = 30 ; ;

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

p = a+b+c = 13+21+30 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-13)(32-21)(32-30) } ; ; T = sqrt{ 13376 } = 115.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 115.65 }{ 13 } = 17.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 115.65 }{ 21 } = 11.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 115.65 }{ 30 } = 7.71 ; ;

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**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 21° 32'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-30**2 }{ 2 * 13 * 30 } ) = 36° 22'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 122° 4'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 115.65 }{ 32 } = 3.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 21° 32'26" } = 17.7 ; ;




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