15 19 20 triangle

Acute scalene triangle.

Sides: a = 15   b = 19   c = 20

Area: T = 134.7699665924
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 45.14991919008° = 45°8'57″ = 0.78880020533 rad
Angle ∠ B = β = 63.89661188627° = 63°53'46″ = 1.11551976534 rad
Angle ∠ C = γ = 70.95546892365° = 70°57'17″ = 1.23883929469 rad

Height: ha = 17.96599554565
Height: hb = 14.17989122025
Height: hc = 13.47699665924

Median: ma = 18.00769431054
Median: mb = 14.90880515159
Median: mc = 13.89224439894

Inradius: r = 4.98988765157
Circumradius: R = 10.57990908257

Vertex coordinates: A[20; 0] B[0; 0] C[6.6; 13.47699665924]
Centroid: CG[8.86766666667; 4.49899888641]
Coordinates of the circumscribed circle: U[10; 3.45221243747]
Coordinates of the inscribed circle: I[8; 4.98988765157]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.8510808099° = 134°51'3″ = 0.78880020533 rad
∠ B' = β' = 116.1043881137° = 116°6'14″ = 1.11551976534 rad
∠ C' = γ' = 109.0455310763° = 109°2'43″ = 1.23883929469 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 = 19 ; ; c = 20 ; ;

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

p = a+b+c = 15+19+20 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-15)(27-19)(27-20) } ; ; T = sqrt{ 18144 } = 134.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.7 }{ 15 } = 17.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.7 }{ 19 } = 14.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.7 }{ 20 } = 13.47 ; ;

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**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 45° 8'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-15**2-20**2 }{ 2 * 15 * 20 } ) = 63° 53'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-15**2-19**2 }{ 2 * 19 * 15 } ) = 70° 57'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.7 }{ 27 } = 4.99 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 45° 8'57" } = 10.58 ; ;




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