12 27 29 triangle

Acute scalene triangle.

Sides: a = 12   b = 27   c = 29

Area: T = 161.8022348561
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 24.41215653884° = 24°24'42″ = 0.42660621916 rad
Angle ∠ B = β = 68.41990070818° = 68°25'8″ = 1.19441369445 rad
Angle ∠ C = γ = 87.16994275297° = 87°10'10″ = 1.52113935175 rad

Height: ha = 26.96770580936
Height: hb = 11.98553591527
Height: hc = 11.15987826594

Median: ma = 27.36878643668
Median: mb = 17.61439149538
Median: mc = 15.04216089565

Inradius: r = 4.75988926047
Circumradius: R = 14.51877126345

Vertex coordinates: A[29; 0] B[0; 0] C[4.41437931034; 11.15987826594]
Centroid: CG[11.13879310345; 3.72195942198]
Coordinates of the circumscribed circle: U[14.5; 0.71769240807]
Coordinates of the inscribed circle: I[7; 4.75988926047]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.5888434612° = 155°35'18″ = 0.42660621916 rad
∠ B' = β' = 111.5810992918° = 111°34'52″ = 1.19441369445 rad
∠ C' = γ' = 92.83105724703° = 92°49'50″ = 1.52113935175 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 = 27 ; ; c = 29 ; ;

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

p = a+b+c = 12+27+29 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-12)(34-27)(34-29) } ; ; T = sqrt{ 26180 } = 161.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 161.8 }{ 12 } = 26.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 161.8 }{ 27 } = 11.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 161.8 }{ 29 } = 11.16 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 27**2+29**2-12**2 }{ 2 * 27 * 29 } ) = 24° 24'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+29**2-27**2 }{ 2 * 12 * 29 } ) = 68° 25'8" ; ; gamma = 180° - alpha - beta = 180° - 24° 24'42" - 68° 25'8" = 87° 10'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 161.8 }{ 34 } = 4.76 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 24° 24'42" } = 14.52 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 29**2 - 12**2 } }{ 2 } = 27.368 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29**2+2 * 12**2 - 27**2 } }{ 2 } = 17.614 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 12**2 - 29**2 } }{ 2 } = 15.042 ; ;
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