19 20 27 triangle

Acute scalene triangle.

Sides: a = 19   b = 20   c = 27

Area: T = 189.8321504235
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 44.67546095644° = 44°40'29″ = 0.78797190289 rad
Angle ∠ B = β = 47.73985577002° = 47°44'19″ = 0.8333195012 rad
Angle ∠ C = γ = 87.58768327354° = 87°35'13″ = 1.52986786126 rad

Height: ha = 19.98222636037
Height: hb = 18.98331504235
Height: hc = 14.06215929063

Median: ma = 21.77772817404
Median: mb = 21.09550231097
Median: mc = 14.08801278403

Inradius: r = 5.75224698253
Circumradius: R = 13.5121982694

Vertex coordinates: A[27; 0] B[0; 0] C[12.77877777778; 14.06215929063]
Centroid: CG[13.25992592593; 4.68771976354]
Coordinates of the circumscribed circle: U[13.5; 0.56989255871]
Coordinates of the inscribed circle: I[13; 5.75224698253]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.3255390436° = 135°19'31″ = 0.78797190289 rad
∠ B' = β' = 132.26114423° = 132°15'41″ = 0.8333195012 rad
∠ C' = γ' = 92.41331672646° = 92°24'47″ = 1.52986786126 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 = 19 ; ; b = 20 ; ; c = 27 ; ;

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

p = a+b+c = 19+20+27 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-19)(33-20)(33-27) } ; ; T = sqrt{ 36036 } = 189.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 189.83 }{ 19 } = 19.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 189.83 }{ 20 } = 18.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 189.83 }{ 27 } = 14.06 ; ;

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{ 19**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 44° 40'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 47° 44'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-19**2-20**2 }{ 2 * 20 * 19 } ) = 87° 35'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 189.83 }{ 33 } = 5.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 44° 40'29" } = 13.51 ; ;




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