13 14 23 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 14   c = 23

Area: T = 81.24403840464
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 30.30547140565° = 30°18'17″ = 0.52989170392 rad
Angle ∠ B = β = 32.9166344351° = 32°54'59″ = 0.57444985866 rad
Angle ∠ C = γ = 116.7798941592° = 116°46'44″ = 2.03881770278 rad

Height: ha = 12.49985206225
Height: hb = 11.60657691495
Height: hc = 7.06443812214

Median: ma = 17.89655301682
Median: mb = 17.32105080757
Median: mc = 7.08987234394

Inradius: r = 3.25496153619
Circumradius: R = 12.8821524531

Vertex coordinates: A[23; 0] B[0; 0] C[10.91330434783; 7.06443812214]
Centroid: CG[11.30443478261; 2.35547937405]
Coordinates of the circumscribed circle: U[11.5; -5.80437637997]
Coordinates of the inscribed circle: I[11; 3.25496153619]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.6955285943° = 149°41'43″ = 0.52989170392 rad
∠ B' = β' = 147.0843655649° = 147°5'1″ = 0.57444985866 rad
∠ C' = γ' = 63.22110584075° = 63°13'16″ = 2.03881770278 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 = 14 ; ; c = 23 ; ;

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

p = a+b+c = 13+14+23 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-13)(25-14)(25-23) } ; ; T = sqrt{ 6600 } = 81.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 81.24 }{ 13 } = 12.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 81.24 }{ 14 } = 11.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 81.24 }{ 23 } = 7.06 ; ;

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**2-14**2-23**2 }{ 2 * 14 * 23 } ) = 30° 18'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 32° 54'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-14**2 }{ 2 * 14 * 13 } ) = 116° 46'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 81.24 }{ 25 } = 3.25 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 30° 18'17" } = 12.88 ; ;




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