14 20 21 triangle

Acute scalene triangle.

Sides: a = 14   b = 20   c = 21

Area: T = 134.533043336
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 39.83881498056° = 39°50'17″ = 0.6955306882 rad
Angle ∠ B = β = 66.23303095773° = 66°13'49″ = 1.15659369667 rad
Angle ∠ C = γ = 73.9321540617° = 73°55'54″ = 1.29903488048 rad

Height: ha = 19.21986333371
Height: hb = 13.4533043336
Height: hc = 12.81224222248

Median: ma = 19.27443352674
Median: mb = 14.78217454991
Median: mc = 13.7022189606

Inradius: r = 4.89220157585
Circumradius: R = 10.92768955974

Vertex coordinates: A[21; 0] B[0; 0] C[5.64328571429; 12.81224222248]
Centroid: CG[8.8810952381; 4.27108074083]
Coordinates of the circumscribed circle: U[10.5; 3.02444086029]
Coordinates of the inscribed circle: I[7.5; 4.89220157585]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.1621850194° = 140°9'43″ = 0.6955306882 rad
∠ B' = β' = 113.7769690423° = 113°46'11″ = 1.15659369667 rad
∠ C' = γ' = 106.0688459383° = 106°4'6″ = 1.29903488048 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 14+20+21 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-14)(27.5-20)(27.5-21) } ; ; T = sqrt{ 18098.44 } = 134.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.53 }{ 14 } = 19.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.53 }{ 20 } = 13.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.53 }{ 21 } = 12.81 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 39° 50'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-14**2-21**2 }{ 2 * 14 * 21 } ) = 66° 13'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-14**2-20**2 }{ 2 * 20 * 14 } ) = 73° 55'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.53 }{ 27.5 } = 4.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 39° 50'17" } = 10.93 ; ;




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