8 28 29 triangle

Acute scalene triangle.

Sides: a = 8   b = 28   c = 29

Area: T = 111.986632729
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 16.01113870121° = 16°41″ = 0.27994514212 rad
Angle ∠ B = β = 74.88438959745° = 74°53'2″ = 1.30769705415 rad
Angle ∠ C = γ = 89.10547170134° = 89°6'17″ = 1.55551706909 rad

Height: ha = 27.99765818226
Height: hb = 7.99990233779
Height: hc = 7.72331949855

Median: ma = 28.22223315833
Median: mb = 16.0165617378
Median: mc = 14.62201915172

Inradius: r = 3.44657331474
Circumradius: R = 14.50217703437

Vertex coordinates: A[29; 0] B[0; 0] C[2.08662068966; 7.72331949855]
Centroid: CG[10.36220689655; 2.57443983285]
Coordinates of the circumscribed circle: U[14.5; 0.22765901616]
Coordinates of the inscribed circle: I[4.5; 3.44657331474]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.9898612988° = 163°59'19″ = 0.27994514212 rad
∠ B' = β' = 105.1166104025° = 105°6'58″ = 1.30769705415 rad
∠ C' = γ' = 90.89552829866° = 90°53'43″ = 1.55551706909 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 = 8 ; ; b = 28 ; ; c = 29 ; ;

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

p = a+b+c = 8+28+29 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-8)(32.5-28)(32.5-29) } ; ; T = sqrt{ 12540.94 } = 111.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 111.99 }{ 8 } = 28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 111.99 }{ 28 } = 8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 111.99 }{ 29 } = 7.72 ; ;

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{ 8**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 16° 41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-8**2-29**2 }{ 2 * 8 * 29 } ) = 74° 53'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-8**2-28**2 }{ 2 * 28 * 8 } ) = 89° 6'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 111.99 }{ 32.5 } = 3.45 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 16° 41" } = 14.5 ; ;




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