21 27 28 triangle

Acute scalene triangle.

Sides: a = 21   b = 27   c = 28

Area: T = 266.5710816107
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 44.84768191535° = 44°50'49″ = 0.78327246533 rad
Angle ∠ B = β = 65.05436961589° = 65°3'13″ = 1.13554011886 rad
Angle ∠ C = γ = 70.09994846876° = 70°5'58″ = 1.22334668118 rad

Height: ha = 25.38876967721
Height: hb = 19.74659863783
Height: hc = 19.04107725791

Median: ma = 25.42114476378
Median: mb = 20.74224685127
Median: mc = 19.72330829233

Inradius: r = 7.01550214765
Circumradius: R = 14.88991017327

Vertex coordinates: A[28; 0] B[0; 0] C[8.85771428571; 19.04107725791]
Centroid: CG[12.28657142857; 6.3476924193]
Coordinates of the circumscribed circle: U[14; 5.06880716656]
Coordinates of the inscribed circle: I[11; 7.01550214765]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.1533180846° = 135°9'11″ = 0.78327246533 rad
∠ B' = β' = 114.9466303841° = 114°56'47″ = 1.13554011886 rad
∠ C' = γ' = 109.9010515312° = 109°54'2″ = 1.22334668118 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 = 21 ; ; b = 27 ; ; c = 28 ; ;

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

p = a+b+c = 21+27+28 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-21)(38-27)(38-28) } ; ; T = sqrt{ 71060 } = 266.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 266.57 }{ 21 } = 25.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 266.57 }{ 27 } = 19.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 266.57 }{ 28 } = 19.04 ; ;

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{ 21**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 44° 50'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 65° 3'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-21**2-27**2 }{ 2 * 27 * 21 } ) = 70° 5'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 266.57 }{ 38 } = 7.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 44° 50'49" } = 14.89 ; ;




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