12 17 20 triangle

Acute scalene triangle.

Sides: a = 12   b = 17   c = 20

Area: T = 101.6665812838
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 36.7299236457° = 36°43'45″ = 0.64110461079 rad
Angle ∠ B = β = 57.91100487437° = 57°54'36″ = 1.01107210206 rad
Angle ∠ C = γ = 85.36107147993° = 85°21'39″ = 1.49898255251 rad

Height: ha = 16.94443021397
Height: hb = 11.96106838633
Height: hc = 10.16765812838

Median: ma = 17.56441680703
Median: mb = 14.13332940251
Median: mc = 10.79435165725

Inradius: r = 4.15496250138
Circumradius: R = 10.03328711445

Vertex coordinates: A[20; 0] B[0; 0] C[6.375; 10.16765812838]
Centroid: CG[8.79216666667; 3.38988604279]
Coordinates of the circumscribed circle: U[10; 0.81114822249]
Coordinates of the inscribed circle: I[7.5; 4.15496250138]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.2710763543° = 143°16'15″ = 0.64110461079 rad
∠ B' = β' = 122.0989951256° = 122°5'24″ = 1.01107210206 rad
∠ C' = γ' = 94.63992852007° = 94°38'21″ = 1.49898255251 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 = 12 ; ; b = 17 ; ; c = 20 ; ;

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

p = a+b+c = 12+17+20 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-12)(24.5-17)(24.5-20) } ; ; T = sqrt{ 10335.94 } = 101.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 101.67 }{ 12 } = 16.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 101.67 }{ 17 } = 11.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 101.67 }{ 20 } = 10.17 ; ;

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{ 12**2-17**2-20**2 }{ 2 * 17 * 20 } ) = 36° 43'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 57° 54'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-12**2-17**2 }{ 2 * 17 * 12 } ) = 85° 21'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 101.67 }{ 24.5 } = 4.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 36° 43'45" } = 10.03 ; ;




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