9 21 22 triangle

Acute scalene triangle.

Sides: a = 9   b = 21   c = 22

Area: T = 94.02112741883
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 24.0187678915° = 24°1'4″ = 0.41991875758 rad
Angle ∠ B = β = 71.75219603276° = 71°45'7″ = 1.25223079525 rad
Angle ∠ C = γ = 84.23303607573° = 84°13'49″ = 1.47700971254 rad

Height: ha = 20.89436164863
Height: hb = 8.95444070656
Height: hc = 8.54773885626

Median: ma = 21.03297408448
Median: mb = 13.12444047484
Median: mc = 11.83221595662

Inradius: r = 3.61662028534
Circumradius: R = 11.05660084297

Vertex coordinates: A[22; 0] B[0; 0] C[2.81881818182; 8.54773885626]
Centroid: CG[8.27327272727; 2.84991295209]
Coordinates of the circumscribed circle: U[11; 1.11114505829]
Coordinates of the inscribed circle: I[5; 3.61662028534]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.9822321085° = 155°58'56″ = 0.41991875758 rad
∠ B' = β' = 108.2488039672° = 108°14'53″ = 1.25223079525 rad
∠ C' = γ' = 95.77696392427° = 95°46'11″ = 1.47700971254 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 = 21 ; ; c = 22 ; ;

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

p = a+b+c = 9+21+22 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-9)(26-21)(26-22) } ; ; T = sqrt{ 8840 } = 94.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 94.02 }{ 9 } = 20.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 94.02 }{ 21 } = 8.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 94.02 }{ 22 } = 8.55 ; ;

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**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 24° 1'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-9**2-22**2 }{ 2 * 9 * 22 } ) = 71° 45'7" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-9**2-21**2 }{ 2 * 21 * 9 } ) = 84° 13'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 94.02 }{ 26 } = 3.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 24° 1'4" } = 11.06 ; ;




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