15 17 19 triangle

Acute scalene triangle.

Sides: a = 15   b = 17   c = 19

Area: T = 121.6277248181
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 48.86604895851° = 48°51'38″ = 0.85327764174 rad
Angle ∠ B = β = 58.59771135386° = 58°35'50″ = 1.02327125634 rad
Angle ∠ C = γ = 72.54223968763° = 72°32'33″ = 1.26661036728 rad

Height: ha = 16.21769664241
Height: hb = 14.30990880213
Height: hc = 12.80328682295

Median: ma = 16.39435963108
Median: mb = 14.85876579581
Median: mc = 12.91331715701

Inradius: r = 4.77696960071
Circumradius: R = 9.95987059489

Vertex coordinates: A[19; 0] B[0; 0] C[7.81657894737; 12.80328682295]
Centroid: CG[8.93985964912; 4.26876227432]
Coordinates of the circumscribed circle: U[9.5; 2.98876117847]
Coordinates of the inscribed circle: I[8.5; 4.77696960071]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.1439510415° = 131°8'22″ = 0.85327764174 rad
∠ B' = β' = 121.4032886461° = 121°24'10″ = 1.02327125634 rad
∠ C' = γ' = 107.4587603124° = 107°27'27″ = 1.26661036728 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 = 17 ; ; c = 19 ; ;

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

p = a+b+c = 15+17+19 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-15)(25.5-17)(25.5-19) } ; ; T = sqrt{ 14793.19 } = 121.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 121.63 }{ 15 } = 16.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 121.63 }{ 17 } = 14.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 121.63 }{ 19 } = 12.8 ; ;

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**2-17**2-19**2 }{ 2 * 17 * 19 } ) = 48° 51'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 58° 35'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-15**2-17**2 }{ 2 * 17 * 15 } ) = 72° 32'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 121.63 }{ 25.5 } = 4.77 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 48° 51'38" } = 9.96 ; ;




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