13 16 22 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 16   c = 22

Area: T = 102.9498712959
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 35.79884595862° = 35°47'54″ = 0.62548009869 rad
Angle ∠ B = β = 46.04879641864° = 46°2'53″ = 0.80436885889 rad
Angle ∠ C = γ = 98.15435762274° = 98°9'13″ = 1.71331030778 rad

Height: ha = 15.83882635322
Height: hb = 12.86985891199
Height: hc = 9.35989739054

Median: ma = 18.10438669902
Median: mb = 16.2021851746
Median: mc = 9.56655632349

Inradius: r = 4.03772044298
Circumradius: R = 11.11223293057

Vertex coordinates: A[22; 0] B[0; 0] C[9.02327272727; 9.35989739054]
Centroid: CG[10.34109090909; 3.12196579685]
Coordinates of the circumscribed circle: U[11; -1.57660274736]
Coordinates of the inscribed circle: I[9.5; 4.03772044298]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.2021540414° = 144°12'6″ = 0.62548009869 rad
∠ B' = β' = 133.9522035814° = 133°57'7″ = 0.80436885889 rad
∠ C' = γ' = 81.84664237726° = 81°50'47″ = 1.71331030778 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 = 16 ; ; c = 22 ; ;

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

p = a+b+c = 13+16+22 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-13)(25.5-16)(25.5-22) } ; ; T = sqrt{ 10598.44 } = 102.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 102.95 }{ 13 } = 15.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 102.95 }{ 16 } = 12.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 102.95 }{ 22 } = 9.36 ; ;

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**2-16**2-22**2 }{ 2 * 16 * 22 } ) = 35° 47'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-13**2-22**2 }{ 2 * 13 * 22 } ) = 46° 2'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-13**2-16**2 }{ 2 * 16 * 13 } ) = 98° 9'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 102.95 }{ 25.5 } = 4.04 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 35° 47'54" } = 11.11 ; ;




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