12 16 19 triangle

Acute scalene triangle.

Sides: a = 12   b = 16   c = 19

Area: T = 95.50435994086
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 38.92657794574° = 38°55'33″ = 0.67993830154 rad
Angle ∠ B = β = 56.90333738225° = 56°54'12″ = 0.99331512287 rad
Angle ∠ C = γ = 84.17108467201° = 84°10'15″ = 1.46990584095 rad

Height: ha = 15.91772665681
Height: hb = 11.93879499261
Height: hc = 10.05330104641

Median: ma = 16.50875740192
Median: mb = 13.73295302177
Median: mc = 10.47661634199

Inradius: r = 4.06439829536
Circumradius: R = 9.54993783025

Vertex coordinates: A[19; 0] B[0; 0] C[6.55326315789; 10.05330104641]
Centroid: CG[8.51875438596; 3.3511003488]
Coordinates of the circumscribed circle: U[9.5; 0.97698587338]
Coordinates of the inscribed circle: I[7.5; 4.06439829536]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.0744220543° = 141°4'27″ = 0.67993830154 rad
∠ B' = β' = 123.0976626178° = 123°5'48″ = 0.99331512287 rad
∠ C' = γ' = 95.82991532799° = 95°49'45″ = 1.46990584095 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 = 12 ; ; b = 16 ; ; c = 19 ; ;

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

p = a+b+c = 12+16+19 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-12)(23.5-16)(23.5-19) } ; ; T = sqrt{ 9120.94 } = 95.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 95.5 }{ 12 } = 15.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 95.5 }{ 16 } = 11.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 95.5 }{ 19 } = 10.05 ; ;

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{ 12**2-16**2-19**2 }{ 2 * 16 * 19 } ) = 38° 55'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-12**2-19**2 }{ 2 * 12 * 19 } ) = 56° 54'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-12**2-16**2 }{ 2 * 16 * 12 } ) = 84° 10'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 95.5 }{ 23.5 } = 4.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 38° 55'33" } = 9.55 ; ;




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