16 21 22 triangle

Acute scalene triangle.

Sides: a = 16   b = 21   c = 22

Area: T = 159.3377495587
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 43.61221194431° = 43°36'44″ = 0.76111750781 rad
Angle ∠ B = β = 64.86773172888° = 64°52'2″ = 1.13221482636 rad
Angle ∠ C = γ = 71.52105632681° = 71°31'14″ = 1.24882693119 rad

Height: ha = 19.91771869484
Height: hb = 15.17549995797
Height: hc = 14.48552268716

Median: ma = 19.96224647777
Median: mb = 16.11767614613
Median: mc = 15.0833103129

Inradius: r = 5.40112710369
Circumradius: R = 11.59880233855

Vertex coordinates: A[22; 0] B[0; 0] C[6.79554545455; 14.48552268716]
Centroid: CG[9.59884848485; 4.82884089572]
Coordinates of the circumscribed circle: U[11; 3.67661591981]
Coordinates of the inscribed circle: I[8.5; 5.40112710369]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.3887880557° = 136°23'16″ = 0.76111750781 rad
∠ B' = β' = 115.1332682711° = 115°7'58″ = 1.13221482636 rad
∠ C' = γ' = 108.4799436732° = 108°28'46″ = 1.24882693119 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 = 16 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 16+21+22 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-16)(29.5-21)(29.5-22) } ; ; T = sqrt{ 25388.44 } = 159.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 159.34 }{ 16 } = 19.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 159.34 }{ 21 } = 15.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 159.34 }{ 22 } = 14.49 ; ;

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{ 16**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 43° 36'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-16**2-22**2 }{ 2 * 16 * 22 } ) = 64° 52'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-16**2-21**2 }{ 2 * 21 * 16 } ) = 71° 31'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 159.34 }{ 29.5 } = 5.4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 43° 36'44" } = 11.6 ; ;




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