5 10 13 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 10   c = 13

Area: T = 22.45499443206
Perimeter: p = 28
Semiperimeter: s = 14

Angle ∠ A = α = 20.20552235834° = 20°12'19″ = 0.35326476776 rad
Angle ∠ B = β = 43.69108952793° = 43°41'27″ = 0.76325499758 rad
Angle ∠ C = γ = 116.1043881137° = 116°6'14″ = 2.02663950002 rad

Height: ha = 8.98799777283
Height: hb = 4.49899888641
Height: hc = 3.45438375878

Median: ma = 11.32547516529
Median: mb = 8.48552813742
Median: mc = 4.5

Inradius: r = 1.60435674515
Circumradius: R = 7.23883253018

Vertex coordinates: A[13; 0] B[0; 0] C[3.61553846154; 3.45438375878]
Centroid: CG[5.53884615385; 1.15112791959]
Coordinates of the circumscribed circle: U[6.5; -3.18548631328]
Coordinates of the inscribed circle: I[4; 1.60435674515]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.7954776417° = 159°47'41″ = 0.35326476776 rad
∠ B' = β' = 136.3099104721° = 136°18'33″ = 0.76325499758 rad
∠ C' = γ' = 63.89661188627° = 63°53'46″ = 2.02663950002 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 = 10 ; ; c = 13 ; ;

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

p = a+b+c = 5+10+13 = 28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28 }{ 2 } = 14 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14 * (14-5)(14-10)(14-13) } ; ; T = sqrt{ 504 } = 22.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.45 }{ 5 } = 8.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.45 }{ 10 } = 4.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.45 }{ 13 } = 3.45 ; ;

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{ 10**2+13**2-5**2 }{ 2 * 10 * 13 } ) = 20° 12'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+13**2-10**2 }{ 2 * 5 * 13 } ) = 43° 41'27" ; ; gamma = 180° - alpha - beta = 180° - 20° 12'19" - 43° 41'27" = 116° 6'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.45 }{ 14 } = 1.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 20° 12'19" } = 7.24 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 13**2 - 5**2 } }{ 2 } = 11.325 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 5**2 - 10**2 } }{ 2 } = 8.485 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 5**2 - 13**2 } }{ 2 } = 4.5 ; ;
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