15 17 22 triangle

Acute scalene triangle.

Sides: a = 15   b = 17   c = 22

Area: T = 127.2799220614
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 42.89334830421° = 42°53'37″ = 0.74986325067 rad
Angle ∠ B = β = 50.47988036414° = 50°28'44″ = 0.8811021326 rad
Angle ∠ C = γ = 86.62877133166° = 86°37'40″ = 1.51219388208 rad

Height: ha = 16.97105627485
Height: hb = 14.97440259545
Height: hc = 11.57108382376

Median: ma = 18.17327818454
Median: mb = 16.88002976164
Median: mc = 11.66219037897

Inradius: r = 4.71440452079
Circumradius: R = 11.01990806735

Vertex coordinates: A[22; 0] B[0; 0] C[9.54554545455; 11.57108382376]
Centroid: CG[10.51551515152; 3.85769460792]
Coordinates of the circumscribed circle: U[11; 0.64881812161]
Coordinates of the inscribed circle: I[10; 4.71440452079]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.1076516958° = 137°6'23″ = 0.74986325067 rad
∠ B' = β' = 129.5211196359° = 129°31'16″ = 0.8811021326 rad
∠ C' = γ' = 93.37222866834° = 93°22'20″ = 1.51219388208 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 = 15 ; ; b = 17 ; ; c = 22 ; ;

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

p = a+b+c = 15+17+22 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-15)(27-17)(27-22) } ; ; T = sqrt{ 16200 } = 127.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127.28 }{ 15 } = 16.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127.28 }{ 17 } = 14.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127.28 }{ 22 } = 11.57 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 17**2+22**2-15**2 }{ 2 * 17 * 22 } ) = 42° 53'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15**2+22**2-17**2 }{ 2 * 15 * 22 } ) = 50° 28'44" ; ; gamma = 180° - alpha - beta = 180° - 42° 53'37" - 50° 28'44" = 86° 37'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127.28 }{ 27 } = 4.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15 }{ 2 * sin 42° 53'37" } = 11.02 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 22**2 - 15**2 } }{ 2 } = 18.173 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 15**2 - 17**2 } }{ 2 } = 16.8 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 15**2 - 22**2 } }{ 2 } = 11.662 ; ;
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