15 16 19 triangle

Acute scalene triangle.

Sides: a = 15   b = 16   c = 19

Area: T = 116.1989500386
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 49.85440523379° = 49°51'15″ = 0.87701173588 rad
Angle ∠ B = β = 54.62334598481° = 54°37'24″ = 0.95333592232 rad
Angle ∠ C = γ = 75.52224878141° = 75°31'21″ = 1.31881160717 rad

Height: ha = 15.49219333848
Height: hb = 14.52436875483
Height: hc = 12.23304737249

Median: ma = 15.88223801743
Median: mb = 15.13327459504
Median: mc = 12.25876506721

Inradius: r = 4.64875800154
Circumradius: R = 9.81215578104

Vertex coordinates: A[19; 0] B[0; 0] C[8.68442105263; 12.23304737249]
Centroid: CG[9.22880701754; 4.0776824575]
Coordinates of the circumscribed circle: U[9.5; 2.45328894526]
Coordinates of the inscribed circle: I[9; 4.64875800154]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.1465947662° = 130°8'45″ = 0.87701173588 rad
∠ B' = β' = 125.3776540152° = 125°22'36″ = 0.95333592232 rad
∠ C' = γ' = 104.4787512186° = 104°28'39″ = 1.31881160717 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 = 15 ; ; b = 16 ; ; c = 19 ; ;

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

p = a+b+c = 15+16+19 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-15)(25-16)(25-19) } ; ; T = sqrt{ 13500 } = 116.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 116.19 }{ 15 } = 15.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 116.19 }{ 16 } = 14.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 116.19 }{ 19 } = 12.23 ; ;

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{ 15**2-16**2-19**2 }{ 2 * 16 * 19 } ) = 49° 51'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 54° 37'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-15**2-16**2 }{ 2 * 16 * 15 } ) = 75° 31'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 116.19 }{ 25 } = 4.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 49° 51'15" } = 9.81 ; ;




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