12 24 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 24   c = 27

Area: T = 143.9822420802
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 26.38443297494° = 26°23'4″ = 0.46604934251 rad
Angle ∠ B = β = 62.7220387264° = 62°43'13″ = 1.09546772659 rad
Angle ∠ C = γ = 90.89552829866° = 90°53'43″ = 1.58664219626 rad

Height: ha = 23.99770701337
Height: hb = 11.99985350668
Height: hc = 10.66553645039

Median: ma = 24.82994180359
Median: mb = 17.10326313765
Median: mc = 13.3322291626

Inradius: r = 4.57108705017
Circumradius: R = 13.5021648251

Vertex coordinates: A[27; 0] B[0; 0] C[5.5; 10.66553645039]
Centroid: CG[10.83333333333; 3.55551215013]
Coordinates of the circumscribed circle: U[13.5; -0.21109632539]
Coordinates of the inscribed circle: I[7.5; 4.57108705017]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.6165670251° = 153°36'56″ = 0.46604934251 rad
∠ B' = β' = 117.2879612736° = 117°16'47″ = 1.09546772659 rad
∠ C' = γ' = 89.10547170134° = 89°6'17″ = 1.58664219626 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 = 12 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 12+24+27 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-12)(31.5-24)(31.5-27) } ; ; T = sqrt{ 20730.94 } = 143.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 143.98 }{ 12 } = 24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 143.98 }{ 24 } = 12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 143.98 }{ 27 } = 10.67 ; ;

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{ 12**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 26° 23'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 62° 43'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-24**2 }{ 2 * 24 * 12 } ) = 90° 53'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 143.98 }{ 31.5 } = 4.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 26° 23'4" } = 13.5 ; ;




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