3 13 15 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 13   c = 15

Area: T = 15.56223744975
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 9.18444977727° = 9°11'4″ = 0.16602997263 rad
Angle ∠ B = β = 43.76217426927° = 43°45'42″ = 0.76437864964 rad
Angle ∠ C = γ = 127.0543759535° = 127°3'14″ = 2.21875064309 rad

Height: ha = 10.37549163317
Height: hb = 2.39442114612
Height: hc = 2.07549832663

Median: ma = 13.9555285737
Median: mb = 8.64658082329
Median: mc = 5.72327615711

Inradius: r = 1.00440241611
Circumradius: R = 9.39876661482

Vertex coordinates: A[15; 0] B[0; 0] C[2.16766666667; 2.07549832663]
Centroid: CG[5.72222222222; 0.69216610888]
Coordinates of the circumscribed circle: U[7.5; -5.66326962688]
Coordinates of the inscribed circle: I[2.5; 1.00440241611]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.8165502227° = 170°48'56″ = 0.16602997263 rad
∠ B' = β' = 136.2388257307° = 136°14'18″ = 0.76437864964 rad
∠ C' = γ' = 52.94662404654° = 52°56'46″ = 2.21875064309 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 = 3 ; ; b = 13 ; ; c = 15 ; ;

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

p = a+b+c = 3+13+15 = 31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-3)(15.5-13)(15.5-15) } ; ; T = sqrt{ 242.19 } = 15.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.56 }{ 3 } = 10.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.56 }{ 13 } = 2.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.56 }{ 15 } = 2.07 ; ;

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{ 3**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 9° 11'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-3**2-15**2 }{ 2 * 3 * 15 } ) = 43° 45'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-3**2-13**2 }{ 2 * 13 * 3 } ) = 127° 3'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.56 }{ 15.5 } = 1 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 9° 11'4" } = 9.4 ; ;




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