12 28 29 triangle

Acute scalene triangle.

Sides: a = 12   b = 28   c = 29

Area: T = 166.5866126373
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 24.22443512236° = 24°13'28″ = 0.4232794688 rad
Angle ∠ B = β = 73.214428532° = 73°12'51″ = 1.27878303383 rad
Angle ∠ C = γ = 82.56113634564° = 82°33'41″ = 1.44109676273 rad

Height: ha = 27.76443543955
Height: hb = 11.89990090267
Height: hc = 11.48986983706

Median: ma = 27.86657495862
Median: mb = 17.21991753577
Median: mc = 15.93295323221

Inradius: r = 4.82985833731
Circumradius: R = 14.62330664764

Vertex coordinates: A[29; 0] B[0; 0] C[3.46655172414; 11.48986983706]
Centroid: CG[10.82218390805; 3.83295661235]
Coordinates of the circumscribed circle: U[14.5; 1.89331648563]
Coordinates of the inscribed circle: I[6.5; 4.82985833731]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.7765648776° = 155°46'32″ = 0.4232794688 rad
∠ B' = β' = 106.786571468° = 106°47'9″ = 1.27878303383 rad
∠ C' = γ' = 97.43986365436° = 97°26'19″ = 1.44109676273 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 = 28 ; ; c = 29 ; ;

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

p = a+b+c = 12+28+29 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-12)(34.5-28)(34.5-29) } ; ; T = sqrt{ 27750.94 } = 166.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 166.59 }{ 12 } = 27.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 166.59 }{ 28 } = 11.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 166.59 }{ 29 } = 11.49 ; ;

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**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 24° 13'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-12**2-29**2 }{ 2 * 12 * 29 } ) = 73° 12'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-12**2-28**2 }{ 2 * 28 * 12 } ) = 82° 33'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 166.59 }{ 34.5 } = 4.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 24° 13'28" } = 14.62 ; ;




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