9 22 23 triangle

Acute scalene triangle.

Sides: a = 9   b = 22   c = 23

Area: T = 98.59900603509
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 22.93548404439° = 22°56'5″ = 0.44002884792 rad
Angle ∠ B = β = 72.28110681266° = 72°16'52″ = 1.26215426257 rad
Angle ∠ C = γ = 84.78440914295° = 84°47'3″ = 1.48797615488 rad

Height: ha = 21.90989023002
Height: hb = 8.96327327592
Height: hc = 8.57330487262

Median: ma = 22.05110770712
Median: mb = 13.56546599663
Median: mc = 12.25876506721

Inradius: r = 3.65114837167
Circumradius: R = 11.54878172541

Vertex coordinates: A[23; 0] B[0; 0] C[2.73991304348; 8.57330487262]
Centroid: CG[8.58797101449; 2.85876829087]
Coordinates of the circumscribed circle: U[11.5; 1.05498015686]
Coordinates of the inscribed circle: I[5; 3.65114837167]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.0655159556° = 157°3'55″ = 0.44002884792 rad
∠ B' = β' = 107.7198931873° = 107°43'8″ = 1.26215426257 rad
∠ C' = γ' = 95.21659085705° = 95°12'57″ = 1.48797615488 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 = 9 ; ; b = 22 ; ; c = 23 ; ;

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

p = a+b+c = 9+22+23 = 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 * (27-9)(27-22)(27-23) } ; ; T = sqrt{ 9720 } = 98.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.59 }{ 9 } = 21.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.59 }{ 22 } = 8.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.59 }{ 23 } = 8.57 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 9**2-22**2-23**2 }{ 2 * 22 * 23 } ) = 22° 56'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-9**2-23**2 }{ 2 * 9 * 23 } ) = 72° 16'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-9**2-22**2 }{ 2 * 22 * 9 } ) = 84° 47'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.59 }{ 27 } = 3.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 22° 56'5" } = 11.55 ; ;




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