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: h_a = 8.98799777283
Height: h_b = 4.49899888641
Height: h_c = 3.45438375878

Median: m_a = 11.32547516529
Median: m_b = 8.48552813742
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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