20 21 28 triangle

Acute scalene triangle.

Sides: a = 20   b = 21   c = 28

Area: T = 209.5165960013
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 45.4550065407° = 45°27' = 0.79332532866 rad
Angle ∠ B = β = 48.44108524081° = 48°26'27″ = 0.8455452367 rad
Angle ∠ C = γ = 86.10990821848° = 86°6'33″ = 1.5032887 rad

Height: ha = 20.95215960013
Height: hb = 19.95439009536
Height: hc = 14.96554257152

Median: ma = 22.63884628453
Median: mb = 21.94988040676
Median: mc = 14.98333240638

Inradius: r = 6.07329263772
Circumradius: R = 14.03223438836

Vertex coordinates: A[28; 0] B[0; 0] C[13.26878571429; 14.96554257152]
Centroid: CG[13.7565952381; 4.98884752384]
Coordinates of the circumscribed circle: U[14; 0.95221947635]
Coordinates of the inscribed circle: I[13.5; 6.07329263772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.5549934593° = 134°33' = 0.79332532866 rad
∠ B' = β' = 131.5599147592° = 131°33'33″ = 0.8455452367 rad
∠ C' = γ' = 93.89109178152° = 93°53'27″ = 1.5032887 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 = 20 ; ; b = 21 ; ; c = 28 ; ;

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

p = a+b+c = 20+21+28 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-20)(34.5-21)(34.5-28) } ; ; T = sqrt{ 43896.94 } = 209.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 209.52 }{ 20 } = 20.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 209.52 }{ 21 } = 19.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 209.52 }{ 28 } = 14.97 ; ;

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{ 20**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 45° 27' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 48° 26'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 86° 6'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 209.52 }{ 34.5 } = 6.07 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 45° 27' } = 14.03 ; ;




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