14 21 22 triangle

Acute scalene triangle.

Sides: a = 14   b = 21   c = 22

Area: T = 141.9366385398
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 37.91114752928° = 37°54'41″ = 0.66216800681 rad
Angle ∠ B = β = 67.17106966747° = 67°10'15″ = 1.17223498178 rad
Angle ∠ C = γ = 74.91878280325° = 74°55'4″ = 1.30875627676 rad

Height: ha = 20.27766264855
Height: hb = 13.51877509903
Height: hc = 12.90333077635

Median: ma = 20.33546994077
Median: mb = 15.15875063912
Median: mc = 14.05334693226

Inradius: r = 4.98802240491
Circumradius: R = 11.3922427639

Vertex coordinates: A[22; 0] B[0; 0] C[5.43218181818; 12.90333077635]
Centroid: CG[9.14439393939; 4.30111025878]
Coordinates of the circumscribed circle: U[11; 2.96443561714]
Coordinates of the inscribed circle: I[7.5; 4.98802240491]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.0898524707° = 142°5'19″ = 0.66216800681 rad
∠ B' = β' = 112.8299303325° = 112°49'45″ = 1.17223498178 rad
∠ C' = γ' = 105.0822171967° = 105°4'56″ = 1.30875627676 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 = 14 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 14+21+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 * (28.5-14)(28.5-21)(28.5-22) } ; ; T = sqrt{ 20145.94 } = 141.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 141.94 }{ 14 } = 20.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 141.94 }{ 21 } = 13.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 141.94 }{ 22 } = 12.9 ; ;

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{ 14**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 37° 54'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-14**2-22**2 }{ 2 * 14 * 22 } ) = 67° 10'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-14**2-21**2 }{ 2 * 21 * 14 } ) = 74° 55'4" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 141.94 }{ 28.5 } = 4.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 37° 54'41" } = 11.39 ; ;




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