20 24 27 triangle

Acute scalene triangle.

Sides: a = 20   b = 24   c = 27

Area: T = 231.9220110167
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 45.70990309713° = 45°42'33″ = 0.79877730883 rad
Angle ∠ B = β = 59.22003887371° = 59°12'1″ = 1.03332417019 rad
Angle ∠ C = γ = 75.09105802916° = 75°5'26″ = 1.31105778633 rad

Height: ha = 23.19220110167
Height: hb = 19.32766758473
Height: hc = 17.17992674198

Median: ma = 23.50553185471
Median: mb = 20.50660966544
Median: mc = 17.486570845

Inradius: r = 6.53329608498
Circumradius: R = 13.97703279619

Vertex coordinates: A[27; 0] B[0; 0] C[10.24107407407; 17.17992674198]
Centroid: CG[12.41435802469; 5.72664224733]
Coordinates of the circumscribed circle: U[13.5; 3.59444489652]
Coordinates of the inscribed circle: I[11.5; 6.53329608498]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.2910969029° = 134°17'27″ = 0.79877730883 rad
∠ B' = β' = 120.8799611263° = 120°47'59″ = 1.03332417019 rad
∠ C' = γ' = 104.9099419708° = 104°54'34″ = 1.31105778633 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 = 24 ; ; c = 27 ; ;

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

p = a+b+c = 20+24+27 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-20)(35.5-24)(35.5-27) } ; ; T = sqrt{ 53786.94 } = 231.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 231.92 }{ 20 } = 23.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 231.92 }{ 24 } = 19.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 231.92 }{ 27 } = 17.18 ; ;

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**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 45° 42'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 59° 12'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-20**2-24**2 }{ 2 * 24 * 20 } ) = 75° 5'26" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 231.92 }{ 35.5 } = 6.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 45° 42'33" } = 13.97 ; ;




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