13 15 27 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 15   c = 27

Area: T = 49.92218138693
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 14.27221207873° = 14°16'20″ = 0.2499095499 rad
Angle ∠ B = β = 16.5266260923° = 16°31'35″ = 0.28884376661 rad
Angle ∠ C = γ = 149.202161829° = 149°12'6″ = 2.60440594885 rad

Height: ha = 7.68802790568
Height: hb = 6.65662418492
Height: hc = 3.69879121385

Median: ma = 20.8510659462
Median: mb = 19.81879211826
Median: mc = 3.84105728739

Inradius: r = 1.81553386862
Circumradius: R = 26.36662294693

Vertex coordinates: A[27; 0] B[0; 0] C[12.4632962963; 3.69879121385]
Centroid: CG[13.15443209877; 1.23326373795]
Coordinates of the circumscribed circle: U[13.5; -22.64879150569]
Coordinates of the inscribed circle: I[12.5; 1.81553386862]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.7287879213° = 165°43'40″ = 0.2499095499 rad
∠ B' = β' = 163.4743739077° = 163°28'25″ = 0.28884376661 rad
∠ C' = γ' = 30.79883817103° = 30°47'54″ = 2.60440594885 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 = 15 ; ; c = 27 ; ;

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

p = a+b+c = 13+15+27 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-13)(27.5-15)(27.5-27) } ; ; T = sqrt{ 2492.19 } = 49.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.92 }{ 13 } = 7.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.92 }{ 15 } = 6.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.92 }{ 27 } = 3.7 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 15**2+27**2-13**2 }{ 2 * 15 * 27 } ) = 14° 16'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+27**2-15**2 }{ 2 * 13 * 27 } ) = 16° 31'35" ; ; gamma = 180° - alpha - beta = 180° - 14° 16'20" - 16° 31'35" = 149° 12'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.92 }{ 27.5 } = 1.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13 }{ 2 * sin 14° 16'20" } = 26.37 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 27**2 - 13**2 } }{ 2 } = 20.851 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 13**2 - 15**2 } }{ 2 } = 19.818 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 13**2 - 27**2 } }{ 2 } = 3.841 ; ;
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