15 18 19 triangle

Acute scalene triangle.

Sides: a = 15   b = 18   c = 19

Area: T = 126.5544336156
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 47.73985577002° = 47°44'19″ = 0.8333195012 rad
Angle ∠ B = β = 62.63655316427° = 62°38'8″ = 1.09331962559 rad
Angle ∠ C = γ = 69.62659106571° = 69°37'33″ = 1.21552013857 rad

Height: ha = 16.87439114875
Height: hb = 14.06215929063
Height: hc = 13.32215090691

Median: ma = 16.91989243157
Median: mb = 14.56602197786
Median: mc = 13.57438719605

Inradius: r = 4.86774744676
Circumradius: R = 10.13439870205

Vertex coordinates: A[19; 0] B[0; 0] C[6.89547368421; 13.32215090691]
Centroid: CG[8.63215789474; 4.4410503023]
Coordinates of the circumscribed circle: U[9.5; 3.52881288146]
Coordinates of the inscribed circle: I[8; 4.86774744676]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.26114423° = 132°15'41″ = 0.8333195012 rad
∠ B' = β' = 117.3644468357° = 117°21'52″ = 1.09331962559 rad
∠ C' = γ' = 110.3744089343° = 110°22'27″ = 1.21552013857 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 = 18 ; ; c = 19 ; ;

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

p = a+b+c = 15+18+19 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-15)(26-18)(26-19) } ; ; T = sqrt{ 16016 } = 126.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 126.55 }{ 15 } = 16.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 126.55 }{ 18 } = 14.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 126.55 }{ 19 } = 13.32 ; ;

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**2-18**2-19**2 }{ 2 * 18 * 19 } ) = 47° 44'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 62° 38'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-15**2-18**2 }{ 2 * 18 * 15 } ) = 69° 37'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 126.55 }{ 26 } = 4.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 47° 44'19" } = 10.13 ; ;




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