10 19 22 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 19   c = 22

Area: T = 94.8265827178
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 26.9822098523° = 26°58'56″ = 0.47109264583 rad
Angle ∠ B = β = 59.54878781287° = 59°32'52″ = 1.03993065359 rad
Angle ∠ C = γ = 93.47700233483° = 93°28'12″ = 1.63113596593 rad

Height: ha = 18.96551654356
Height: hb = 9.98216660187
Height: hc = 8.62105297435

Median: ma = 19.93774020374
Median: mb = 14.20438727113
Median: mc = 10.46442247682

Inradius: r = 3.71986598893
Circumradius: R = 11.02202044221

Vertex coordinates: A[22; 0] B[0; 0] C[5.06881818182; 8.62105297435]
Centroid: CG[9.02327272727; 2.87435099145]
Coordinates of the circumscribed circle: U[11; -0.66770123729]
Coordinates of the inscribed circle: I[6.5; 3.71986598893]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.0187901477° = 153°1'4″ = 0.47109264583 rad
∠ B' = β' = 120.4522121871° = 120°27'8″ = 1.03993065359 rad
∠ C' = γ' = 86.53299766517° = 86°31'48″ = 1.63113596593 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 = 10 ; ; b = 19 ; ; c = 22 ; ;

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

p = a+b+c = 10+19+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 * (25.5-10)(25.5-19)(25.5-22) } ; ; T = sqrt{ 8991.94 } = 94.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 94.83 }{ 10 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 94.83 }{ 19 } = 9.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 94.83 }{ 22 } = 8.62 ; ;

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{ 10**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 26° 58'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-10**2-22**2 }{ 2 * 10 * 22 } ) = 59° 32'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-10**2-19**2 }{ 2 * 19 * 10 } ) = 93° 28'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 94.83 }{ 25.5 } = 3.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 26° 58'56" } = 11.02 ; ;




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