20 22 27 triangle

Acute scalene triangle.

Sides: a = 20   b = 22   c = 27

Area: T = 216.566047077
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 46.81660061647° = 46°48'58″ = 0.81770934502 rad
Angle ∠ B = β = 53.32987880616° = 53°19'44″ = 0.93107629378 rad
Angle ∠ C = γ = 79.85552057737° = 79°51'19″ = 1.39437362656 rad

Height: ha = 21.6566047077
Height: hb = 19.68773155245
Height: hc = 16.04215163533

Median: ma = 22.50655548699
Median: mb = 21.05994396887
Median: mc = 16.11767614613

Inradius: r = 6.27771150948
Circumradius: R = 13.71444142209

Vertex coordinates: A[27; 0] B[0; 0] C[11.94444444444; 16.04215163533]
Centroid: CG[12.98114814815; 5.34771721178]
Coordinates of the circumscribed circle: U[13.5; 2.41656070503]
Coordinates of the inscribed circle: I[12.5; 6.27771150948]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.1843993835° = 133°11'2″ = 0.81770934502 rad
∠ B' = β' = 126.6711211938° = 126°40'16″ = 0.93107629378 rad
∠ C' = γ' = 100.1454794226° = 100°8'41″ = 1.39437362656 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 = 22 ; ; c = 27 ; ;

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

p = a+b+c = 20+22+27 = 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)(34.5-22)(34.5-27) } ; ; T = sqrt{ 46898.44 } = 216.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 216.56 }{ 20 } = 21.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 216.56 }{ 22 } = 19.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 216.56 }{ 27 } = 16.04 ; ;

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{ 20**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 46° 48'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 53° 19'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-20**2-22**2 }{ 2 * 22 * 20 } ) = 79° 51'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 216.56 }{ 34.5 } = 6.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 46° 48'58" } = 13.71 ; ;




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