15 17 20 triangle

Acute scalene triangle.

Sides: a = 15   b = 17   c = 20

Area: T = 124.2743891063
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 46.97222150901° = 46°58'20″ = 0.82198198103 rad
Angle ∠ B = β = 55.94442022574° = 55°56'39″ = 0.97664105268 rad
Angle ∠ C = γ = 77.08435826525° = 77°5'1″ = 1.34553623165 rad

Height: ha = 16.57698521418
Height: hb = 14.62204577721
Height: hc = 12.42773891063

Median: ma = 16.97879268463
Median: mb = 15.5
Median: mc = 12.53299640861

Inradius: r = 4.78797650409
Circumradius: R = 10.26595966787

Vertex coordinates: A[20; 0] B[0; 0] C[8.4; 12.42773891063]
Centroid: CG[9.46766666667; 4.14224630354]
Coordinates of the circumscribed circle: U[10; 2.29333216105]
Coordinates of the inscribed circle: I[9; 4.78797650409]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.028778491° = 133°1'40″ = 0.82198198103 rad
∠ B' = β' = 124.0565797743° = 124°3'21″ = 0.97664105268 rad
∠ C' = γ' = 102.9166417347° = 102°54'59″ = 1.34553623165 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 = 20 ; ;

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

p = a+b+c = 15+17+20 = 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)(26-17)(26-20) } ; ; T = sqrt{ 15444 } = 124.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 124.27 }{ 15 } = 16.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 124.27 }{ 17 } = 14.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 124.27 }{ 20 } = 12.43 ; ;

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**2-20**2 }{ 2 * 17 * 20 } ) = 46° 58'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-15**2-20**2 }{ 2 * 15 * 20 } ) = 55° 56'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-15**2-17**2 }{ 2 * 17 * 15 } ) = 77° 5'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 124.27 }{ 26 } = 4.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 46° 58'20" } = 10.26 ; ;




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