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: h_a = 13.35875655047
Height: h_b = 12.4033453683
Height: h_c = 10.85330219726

Median: m_a = 13.55554417117
Median: m_b = 12.78767118525
Median: m_c = 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

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.

$a = 13 ; ; b = 14 ; ; c = 16 ; ;$

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

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

6. Inradius

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

7. Circumradius

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

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