12 20 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 20   c = 27

Area: T = 110.729911767
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 24.21216095233° = 24°12'42″ = 0.42325723034 rad
Angle ∠ B = β = 43.11987806049° = 43°7'8″ = 0.7532564691 rad
Angle ∠ C = γ = 112.6769609872° = 112°40'11″ = 1.96664556592 rad

Height: ha = 18.4554852945
Height: hb = 11.0732911767
Height: hc = 8.20221568645

Median: ma = 22.98991278651
Median: mb = 18.34439363278
Median: mc = 9.47436476607

Inradius: r = 3.75435294125
Circumradius: R = 14.63302981012

Vertex coordinates: A[27; 0] B[0; 0] C[8.75992592593; 8.20221568645]
Centroid: CG[11.92197530864; 2.73440522882]
Coordinates of the circumscribed circle: U[13.5; -5.63987607265]
Coordinates of the inscribed circle: I[9.5; 3.75435294125]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.7888390477° = 155°47'18″ = 0.42325723034 rad
∠ B' = β' = 136.8811219395° = 136°52'52″ = 0.7532564691 rad
∠ C' = γ' = 67.33303901282° = 67°19'49″ = 1.96664556592 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 = 20 ; ; c = 27 ; ;

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

p = a+b+c = 12+20+27 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-12)(29.5-20)(29.5-27) } ; ; T = sqrt{ 12260.94 } = 110.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110.73 }{ 12 } = 18.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110.73 }{ 20 } = 11.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110.73 }{ 27 } = 8.2 ; ;

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**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 24° 12'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 43° 7'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-20**2 }{ 2 * 20 * 12 } ) = 112° 40'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110.73 }{ 29.5 } = 3.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 24° 12'42" } = 14.63 ; ;




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