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: h_a = 26.96770580936
Height: h_b = 11.98553591527
Height: h_c = 11.15987826594

Median: m_a = 27.36878643668
Median: m_b = 17.61439149538
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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