13 14 16 triangle

Acute scalene triangle.

Sides: a = 13   b = 14   c = 16

Area: T = 86.82441757807
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 50.82546068485° = 50°49'29″ = 0.88770567305 rad
Angle ∠ B = β = 56.65999979997° = 56°36' = 0.98878563217 rad
Angle ∠ C = γ = 72.57553951518° = 72°34'31″ = 1.26766796013 rad

Height: ha = 13.35875655047
Height: hb = 12.4033453683
Height: hc = 10.85330219726

Median: ma = 13.55554417117
Median: mb = 12.78767118525
Median: mc = 10.88657705285

Inradius: r = 4.03883337572
Circumradius: R = 8.38547614268

Vertex coordinates: A[16; 0] B[0; 0] C[7.156625; 10.85330219726]
Centroid: CG[7.719875; 3.61876739909]
Coordinates of the circumscribed circle: U[8; 2.51108214163]
Coordinates of the inscribed circle: I[7.5; 4.03883337572]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.1755393151° = 129°10'31″ = 0.88770567305 rad
∠ B' = β' = 123.4400002° = 123°24' = 0.98878563217 rad
∠ C' = γ' = 107.4254604848° = 107°25'29″ = 1.26766796013 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 = 14 ; ; c = 16 ; ;

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

p = a+b+c = 13+14+16 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-13)(21.5-14)(21.5-16) } ; ; T = sqrt{ 7538.44 } = 86.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.82 }{ 13 } = 13.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.82 }{ 14 } = 12.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.82 }{ 16 } = 10.85 ; ;

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{ 14**2+16**2-13**2 }{ 2 * 14 * 16 } ) = 50° 49'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+16**2-14**2 }{ 2 * 13 * 16 } ) = 56° 36' ; ; gamma = 180° - alpha - beta = 180° - 50° 49'29" - 56° 36' = 72° 34'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.82 }{ 21.5 } = 4.04 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13 }{ 2 * sin 50° 49'29" } = 8.38 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 16**2 - 13**2 } }{ 2 } = 13.555 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 13**2 - 14**2 } }{ 2 } = 12.787 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 13**2 - 16**2 } }{ 2 } = 10.886 ; ;
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