16 19 21 triangle

Acute scalene triangle.

Sides: a = 16   b = 19   c = 21

Area: T = 145.4922267836
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 46.82664488927° = 46°49'35″ = 0.81772757102 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 73.17435511073° = 73°10'25″ = 1.27771193922 rad

Height: ha = 18.18765334795
Height: hb = 15.31549755617
Height: hc = 13.85664064606

Median: ma = 18.35875597507
Median: mb = 16.077015868
Median: mc = 14.08801278403

Inradius: r = 5.19661524227
Circumradius: R = 10.97696551146

Vertex coordinates: A[21; 0] B[0; 0] C[8; 13.85664064606]
Centroid: CG[9.66766666667; 4.61988021535]
Coordinates of the circumscribed circle: U[10.5; 3.17554264805]
Coordinates of the inscribed circle: I[9; 5.19661524227]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.1743551107° = 133°10'25″ = 0.81772757102 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 106.8266448893° = 106°49'35″ = 1.27771193922 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 = 19 ; ; c = 21 ; ;

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

p = a+b+c = 16+19+21 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-16)(28-19)(28-21) } ; ; T = sqrt{ 21168 } = 145.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 145.49 }{ 16 } = 18.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 145.49 }{ 19 } = 15.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 145.49 }{ 21 } = 13.86 ; ;

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**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 46° 49'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-16**2-19**2 }{ 2 * 19 * 16 } ) = 73° 10'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 145.49 }{ 28 } = 5.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 46° 49'35" } = 10.97 ; ;




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