17 19 22 triangle

Acute scalene triangle.

Sides: a = 17   b = 19   c = 22

Area: T = 156.0776904121
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 48.31221683473° = 48°18'44″ = 0.84332064064 rad
Angle ∠ B = β = 56.57879395233° = 56°34'41″ = 0.98774713287 rad
Angle ∠ C = γ = 75.11098921294° = 75°6'36″ = 1.31109149185 rad

Height: ha = 18.36219887201
Height: hb = 16.42991478022
Height: hc = 14.18988094655

Median: ma = 18.71549672722
Median: mb = 17.21219144781
Median: mc = 14.28328568571

Inradius: r = 5.38219622111
Circumradius: R = 11.38222093666

Vertex coordinates: A[22; 0] B[0; 0] C[9.36436363636; 14.18988094655]
Centroid: CG[10.45545454545; 4.73296031552]
Coordinates of the circumscribed circle: U[11; 2.92548401778]
Coordinates of the inscribed circle: I[10; 5.38219622111]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.6887831653° = 131°41'16″ = 0.84332064064 rad
∠ B' = β' = 123.4222060477° = 123°25'19″ = 0.98774713287 rad
∠ C' = γ' = 104.8990107871° = 104°53'24″ = 1.31109149185 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 = 17 ; ; b = 19 ; ; c = 22 ; ;

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

p = a+b+c = 17+19+22 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-17)(29-19)(29-22) } ; ; T = sqrt{ 24360 } = 156.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 156.08 }{ 17 } = 18.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 156.08 }{ 19 } = 16.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 156.08 }{ 22 } = 14.19 ; ;

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{ 17**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 48° 18'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-17**2-22**2 }{ 2 * 17 * 22 } ) = 56° 34'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-17**2-19**2 }{ 2 * 19 * 17 } ) = 75° 6'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 156.08 }{ 29 } = 5.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 48° 18'44" } = 11.38 ; ;




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