13 19 28 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 19   c = 28

Area: T = 105.9254501415
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 23.46664751452° = 23°27'59″ = 0.41095672551 rad
Angle ∠ B = β = 35.59215293953° = 35°35'30″ = 0.62111893738 rad
Angle ∠ C = γ = 120.9421995459° = 120°56'31″ = 2.11108360247 rad

Height: ha = 16.29660771408
Height: hb = 11.15499475174
Height: hc = 7.56660358154

Median: ma = 23.02771578793
Median: mb = 19.6533244007
Median: mc = 8.30766238629

Inradius: r = 3.53108167138
Circumradius: R = 16.32329467867

Vertex coordinates: A[28; 0] B[0; 0] C[10.57114285714; 7.56660358154]
Centroid: CG[12.85771428571; 2.52220119385]
Coordinates of the circumscribed circle: U[14; -8.39327702101]
Coordinates of the inscribed circle: I[11; 3.53108167138]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.5343524855° = 156°32'1″ = 0.41095672551 rad
∠ B' = β' = 144.4088470605° = 144°24'30″ = 0.62111893738 rad
∠ C' = γ' = 59.05880045405° = 59°3'29″ = 2.11108360247 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 = 28 ; ;

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

p = a+b+c = 13+19+28 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-13)(30-19)(30-28) } ; ; T = sqrt{ 11220 } = 105.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 105.92 }{ 13 } = 16.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 105.92 }{ 19 } = 11.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 105.92 }{ 28 } = 7.57 ; ;

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**2-28**2 }{ 2 * 19 * 28 } ) = 23° 27'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-28**2 }{ 2 * 13 * 28 } ) = 35° 35'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 120° 56'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 105.92 }{ 30 } = 3.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 23° 27'59" } = 16.32 ; ;




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