5 13 15 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 13   c = 15

Area: T = 31.56224381188
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 18.88878819509° = 18°53'16″ = 0.33296557288 rad
Angle ∠ B = β = 57.31663611537° = 57°18'59″ = 11.0003592174 rad
Angle ∠ C = γ = 103.7965756895° = 103°47'45″ = 1.81215777074 rad

Height: ha = 12.62549752475
Height: hb = 4.85657597106
Height: hc = 4.20883250825

Median: ma = 13.81112273169
Median: mb = 9.09767026993
Median: mc = 6.38435726674

Inradius: r = 1.91328750375
Circumradius: R = 7.72327874185

Vertex coordinates: A[15; 0] B[0; 0] C[2.7; 4.20883250825]
Centroid: CG[5.9; 1.40327750275]
Coordinates of the circumscribed circle: U[7.5; -1.8421587769]
Coordinates of the inscribed circle: I[3.5; 1.91328750375]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.1122118049° = 161°6'44″ = 0.33296557288 rad
∠ B' = β' = 122.6843638846° = 122°41'1″ = 11.0003592174 rad
∠ C' = γ' = 76.20442431047° = 76°12'15″ = 1.81215777074 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 = 5 ; ; b = 13 ; ; c = 15 ; ;

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

p = a+b+c = 5+13+15 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-5)(16.5-13)(16.5-15) } ; ; T = sqrt{ 996.19 } = 31.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.56 }{ 5 } = 12.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.56 }{ 13 } = 4.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.56 }{ 15 } = 4.21 ; ;

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{ 5**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 18° 53'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-5**2-15**2 }{ 2 * 5 * 15 } ) = 57° 18'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-5**2-13**2 }{ 2 * 13 * 5 } ) = 103° 47'45" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.56 }{ 16.5 } = 1.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 18° 53'16" } = 7.72 ; ;




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