14 17 21 triangle

Acute scalene triangle.

Sides: a = 14   b = 17   c = 21

Area: T = 118.4910505949
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 41.59112771434° = 41°35'29″ = 0.72659047263 rad
Angle ∠ B = β = 53.71325429149° = 53°42'45″ = 0.93774607235 rad
Angle ∠ C = γ = 84.69661799417° = 84°41'46″ = 1.47882272038 rad

Height: ha = 16.92772151355
Height: hb = 13.94400595234
Height: hc = 11.28548100904

Median: ma = 17.77663888346
Median: mb = 15.69223548265
Median: mc = 11.5

Inradius: r = 4.55773271519
Circumradius: R = 10.54551486598

Vertex coordinates: A[21; 0] B[0; 0] C[8.28657142857; 11.28548100904]
Centroid: CG[9.76219047619; 3.76216033635]
Coordinates of the circumscribed circle: U[10.5; 0.97547616408]
Coordinates of the inscribed circle: I[9; 4.55773271519]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.4098722857° = 138°24'31″ = 0.72659047263 rad
∠ B' = β' = 126.2877457085° = 126°17'15″ = 0.93774607235 rad
∠ C' = γ' = 95.30438200583° = 95°18'14″ = 1.47882272038 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 = 14 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 14+17+21 = 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 * (26-14)(26-17)(26-21) } ; ; T = sqrt{ 14040 } = 118.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 118.49 }{ 14 } = 16.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 118.49 }{ 17 } = 13.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 118.49 }{ 21 } = 11.28 ; ;

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{ 14**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 41° 35'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-14**2-21**2 }{ 2 * 14 * 21 } ) = 53° 42'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-14**2-17**2 }{ 2 * 17 * 14 } ) = 84° 41'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 118.49 }{ 26 } = 4.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 41° 35'29" } = 10.55 ; ;




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