13 18 23 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 18   c = 23

Area: T = 116.6533332571
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 34.30111527568° = 34°18'4″ = 0.59986680528 rad
Angle ∠ B = β = 51.28771214574° = 51°17'14″ = 0.89551291333 rad
Angle ∠ C = γ = 94.41217257858° = 94°24'42″ = 1.64877954675 rad

Height: ha = 17.94766665494
Height: hb = 12.96114813968
Height: hc = 10.14437680497

Median: ma = 19.60222957839
Median: mb = 16.37107055437
Median: mc = 10.68987791632

Inradius: r = 4.32204937989
Circumradius: R = 11.53441754097

Vertex coordinates: A[23; 0] B[0; 0] C[8.13304347826; 10.14437680497]
Centroid: CG[10.37768115942; 3.38112560166]
Coordinates of the circumscribed circle: U[11.5; -0.88772442623]
Coordinates of the inscribed circle: I[9; 4.32204937989]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.6998847243° = 145°41'56″ = 0.59986680528 rad
∠ B' = β' = 128.7132878543° = 128°42'46″ = 0.89551291333 rad
∠ C' = γ' = 85.58882742142° = 85°35'18″ = 1.64877954675 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 = 18 ; ; c = 23 ; ;

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

p = a+b+c = 13+18+23 = 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)(27-18)(27-23) } ; ; T = sqrt{ 13608 } = 116.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 116.65 }{ 13 } = 17.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 116.65 }{ 18 } = 12.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 116.65 }{ 23 } = 10.14 ; ;

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**2-18**2-23**2 }{ 2 * 18 * 23 } ) = 34° 18'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 51° 17'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-18**2 }{ 2 * 18 * 13 } ) = 94° 24'42" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 116.65 }{ 27 } = 4.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 34° 18'4" } = 11.53 ; ;




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