17 19 20 triangle

Acute scalene triangle.

Sides: a = 17   b = 19   c = 20

Area: T = 148.9166083752
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 51.60769559808° = 51°36'25″ = 0.90107112988 rad
Angle ∠ B = β = 61.16108105994° = 61°9'39″ = 1.06774575181 rad
Angle ∠ C = γ = 67.23222334198° = 67°13'56″ = 1.17334238366 rad

Height: ha = 17.52195392649
Height: hb = 15.6755377237
Height: hc = 14.89216083752

Median: ma = 17.55770498661
Median: mb = 15.94552187191
Median: mc = 15

Inradius: r = 5.31884315626
Circumradius: R = 10.84550340575

Vertex coordinates: A[20; 0] B[0; 0] C[8.2; 14.89216083752]
Centroid: CG[9.4; 4.96438694584]
Coordinates of the circumscribed circle: U[10; 4.19769946043]
Coordinates of the inscribed circle: I[9; 5.31884315626]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.3933044019° = 128°23'35″ = 0.90107112988 rad
∠ B' = β' = 118.8399189401° = 118°50'21″ = 1.06774575181 rad
∠ C' = γ' = 112.768776658° = 112°46'4″ = 1.17334238366 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 = 17 ; ; b = 19 ; ; c = 20 ; ;

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

p = a+b+c = 17+19+20 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-17)(28-19)(28-20) } ; ; T = sqrt{ 22176 } = 148.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 148.92 }{ 17 } = 17.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 148.92 }{ 19 } = 15.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 148.92 }{ 20 } = 14.89 ; ;

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{ 17**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 51° 36'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-17**2-20**2 }{ 2 * 17 * 20 } ) = 61° 9'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-17**2-19**2 }{ 2 * 19 * 17 } ) = 67° 13'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 148.92 }{ 28 } = 5.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 51° 36'25" } = 10.85 ; ;




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