13 22 22 triangle

Acute isosceles triangle.

Sides: a = 13   b = 22   c = 22

Area: T = 136.616602212
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 34.37695918022° = 34°22'11″ = 0.65998625395 rad
Angle ∠ B = β = 72.81552040989° = 72°48'55″ = 1.2710865057 rad
Angle ∠ C = γ = 72.81552040989° = 72°48'55″ = 1.2710865057 rad

Height: ha = 21.0187849557
Height: hb = 12.42196383746
Height: hc = 12.42196383746

Median: ma = 21.0187849557
Median: mb = 14.33552711868
Median: mc = 14.33552711868

Inradius: r = 4.79435446358
Circumradius: R = 11.51440228473

Vertex coordinates: A[22; 0] B[0; 0] C[3.84109090909; 12.42196383746]
Centroid: CG[8.61436363636; 4.14398794582]
Coordinates of the circumscribed circle: U[11; 3.40218703867]
Coordinates of the inscribed circle: I[6.5; 4.79435446358]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.6330408198° = 145°37'49″ = 0.65998625395 rad
∠ B' = β' = 107.1854795901° = 107°11'5″ = 1.2710865057 rad
∠ C' = γ' = 107.1854795901° = 107°11'5″ = 1.2710865057 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 = 13 ; ; b = 22 ; ; c = 22 ; ;

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

p = a+b+c = 13+22+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 * (28.5-13)(28.5-22)(28.5-22) } ; ; T = sqrt{ 18663.94 } = 136.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 136.62 }{ 13 } = 21.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 136.62 }{ 22 } = 12.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 136.62 }{ 22 } = 12.42 ; ;

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{ 13**2-22**2-22**2 }{ 2 * 22 * 22 } ) = 34° 22'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-13**2-22**2 }{ 2 * 13 * 22 } ) = 72° 48'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-13**2-22**2 }{ 2 * 22 * 13 } ) = 72° 48'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 136.62 }{ 28.5 } = 4.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 34° 22'11" } = 11.51 ; ;




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