19 21 27 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 27

Area: T = 198.6643503191
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 44.4877456403° = 44°29'15″ = 0.77664525901 rad
Angle ∠ B = β = 50.76112247363° = 50°45'40″ = 0.8865950504 rad
Angle ∠ C = γ = 84.75113188607° = 84°45'5″ = 1.47991895595 rad

Height: ha = 20.91219477043
Height: hb = 18.92203336372
Height: hc = 14.71658150512

Median: ma = 22.24329764195
Median: mb = 20.8510659462
Median: mc = 14.79901994577

Inradius: r = 5.93302538266
Circumradius: R = 13.55768433897

Vertex coordinates: A[27; 0] B[0; 0] C[12.01985185185; 14.71658150512]
Centroid: CG[13.00661728395; 4.90552716837]
Coordinates of the circumscribed circle: U[13.5; 1.24401623652]
Coordinates of the inscribed circle: I[12.5; 5.93302538266]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5132543597° = 135°30'45″ = 0.77664525901 rad
∠ B' = β' = 129.2398775264° = 129°14'20″ = 0.8865950504 rad
∠ C' = γ' = 95.24986811393° = 95°14'55″ = 1.47991895595 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 = 19 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 19+21+27 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-19)(33.5-21)(33.5-27) } ; ; T = sqrt{ 39467.19 } = 198.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 198.66 }{ 19 } = 20.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 198.66 }{ 21 } = 18.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 198.66 }{ 27 } = 14.72 ; ;

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{ 19**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 44° 29'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 50° 45'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 84° 45'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 198.66 }{ 33.5 } = 5.93 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 44° 29'15" } = 13.56 ; ;




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