13 16 19 triangle

Acute scalene triangle.

Sides: a = 13   b = 16   c = 19

Area: T = 102.7621860629
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 42.5376898363° = 42°32'13″ = 0.742240893 rad
Angle ∠ B = β = 56.31329847354° = 56°18'47″ = 0.98328469953 rad
Angle ∠ C = γ = 81.15501169016° = 81°9' = 1.41663367283 rad

Height: ha = 15.81095170199
Height: hb = 12.84552325787
Height: hc = 10.8177037961

Median: ma = 16.31771688721
Median: mb = 14.17774468788
Median: mc = 11.05766721937

Inradius: r = 4.28217441929
Circumradius: R = 9.61444619604

Vertex coordinates: A[19; 0] B[0; 0] C[7.21105263158; 10.8177037961]
Centroid: CG[8.73768421053; 3.60656793203]
Coordinates of the circumscribed circle: U[9.5; 1.47991479939]
Coordinates of the inscribed circle: I[8; 4.28217441929]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.4633101637° = 137°27'47″ = 0.742240893 rad
∠ B' = β' = 123.6877015265° = 123°41'13″ = 0.98328469953 rad
∠ C' = γ' = 98.85498830984° = 98°51' = 1.41663367283 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 = 13 ; ; b = 16 ; ; c = 19 ; ;

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

p = a+b+c = 13+16+19 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-13)(24-16)(24-19) } ; ; T = sqrt{ 10560 } = 102.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 102.76 }{ 13 } = 15.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 102.76 }{ 16 } = 12.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 102.76 }{ 19 } = 10.82 ; ;

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{ 13**2-16**2-19**2 }{ 2 * 16 * 19 } ) = 42° 32'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-13**2-19**2 }{ 2 * 13 * 19 } ) = 56° 18'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-13**2-16**2 }{ 2 * 16 * 13 } ) = 81° 9' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 102.76 }{ 24 } = 4.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 42° 32'13" } = 9.61 ; ;




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