19 21 22 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 22

Area: T = 182.9755408184
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 52.38223203336° = 52°22'56″ = 0.91442439597 rad
Angle ∠ B = β = 61.10218844875° = 61°6'7″ = 1.06664290635 rad
Angle ∠ C = γ = 66.51657951789° = 66°30'57″ = 1.16109196305 rad

Height: ha = 19.26105692825
Height: hb = 17.42662293508
Height: hc = 16.63441280167

Median: ma = 19.29437813816
Median: mb = 17.67105970471
Median: mc = 16.73332005307

Inradius: r = 5.90224325221
Circumradius: R = 11.99334149719

Vertex coordinates: A[22; 0] B[0; 0] C[9.18218181818; 16.63441280167]
Centroid: CG[10.39439393939; 5.54547093389]
Coordinates of the circumscribed circle: U[11; 4.77993307783]
Coordinates of the inscribed circle: I[10; 5.90224325221]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.6187679666° = 127°37'4″ = 0.91442439597 rad
∠ B' = β' = 118.8988115512° = 118°53'53″ = 1.06664290635 rad
∠ C' = γ' = 113.4844204821° = 113°29'3″ = 1.16109196305 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 = 19 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 19+21+22 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-19)(31-21)(31-22) } ; ; T = sqrt{ 33480 } = 182.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 182.98 }{ 19 } = 19.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 182.98 }{ 21 } = 17.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 182.98 }{ 22 } = 16.63 ; ;

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{ 19**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 52° 22'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 61° 6'7" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 66° 30'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 182.98 }{ 31 } = 5.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 52° 22'56" } = 11.99 ; ;




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