16 19 22 triangle

Acute scalene triangle.

Sides: a = 16   b = 19   c = 22

Area: T = 148.3198702462
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 45.20771662976° = 45°12'26″ = 0.78990138974 rad
Angle ∠ B = β = 57.42884587954° = 57°25'42″ = 1.00223156903 rad
Angle ∠ C = γ = 77.3644374907° = 77°21'52″ = 1.35502630659 rad

Height: ha = 18.54398378077
Height: hb = 15.6122494996
Height: hc = 13.48435184056

Median: ma = 18.93440962288
Median: mb = 16.72657286837
Median: mc = 13.69330639376

Inradius: r = 5.20441649987
Circumradius: R = 11.27330220279

Vertex coordinates: A[22; 0] B[0; 0] C[8.61436363636; 13.48435184056]
Centroid: CG[10.20545454545; 4.49545061352]
Coordinates of the circumscribed circle: U[11; 2.46659735686]
Coordinates of the inscribed circle: I[9.5; 5.20441649987]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.7932833702° = 134°47'34″ = 0.78990138974 rad
∠ B' = β' = 122.5721541205° = 122°34'18″ = 1.00223156903 rad
∠ C' = γ' = 102.6365625093° = 102°38'8″ = 1.35502630659 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 = 16 ; ; b = 19 ; ; c = 22 ; ;

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

p = a+b+c = 16+19+22 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-16)(28.5-19)(28.5-22) } ; ; T = sqrt{ 21998.44 } = 148.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 148.32 }{ 16 } = 18.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 148.32 }{ 19 } = 15.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 148.32 }{ 22 } = 13.48 ; ;

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{ 16**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 45° 12'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-16**2-22**2 }{ 2 * 16 * 22 } ) = 57° 25'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-16**2-19**2 }{ 2 * 19 * 16 } ) = 77° 21'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 148.32 }{ 28.5 } = 5.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 45° 12'26" } = 11.27 ; ;




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