8 9 13 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 9   c = 13

Area: T = 35.49664786986
Perimeter: p = 30
Semiperimeter: s = 15

Angle ∠ A = α = 37.35768519729° = 37°21'25″ = 0.65220000651 rad
Angle ∠ B = β = 43.04990798002° = 43°2'57″ = 0.75113481825 rad
Angle ∠ C = γ = 99.59440682269° = 99°35'39″ = 1.7388244406 rad

Height: ha = 8.87441196746
Height: hb = 7.88881063775
Height: hc = 5.46109967229

Median: ma = 10.44403065089
Median: mb = 9.81107084352
Median: mc = 5.5

Inradius: r = 2.36664319132
Circumradius: R = 6.59222031869

Vertex coordinates: A[13; 0] B[0; 0] C[5.84661538462; 5.46109967229]
Centroid: CG[6.28220512821; 1.8220332241]
Coordinates of the circumscribed circle: U[6.5; -1.09987005311]
Coordinates of the inscribed circle: I[6; 2.36664319132]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.6433148027° = 142°38'35″ = 0.65220000651 rad
∠ B' = β' = 136.95109202° = 136°57'3″ = 0.75113481825 rad
∠ C' = γ' = 80.40659317731° = 80°24'21″ = 1.7388244406 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 = 9 ; ; c = 13 ; ;

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

p = a+b+c = 8+9+13 = 30 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30 }{ 2 } = 15 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15 * (15-8)(15-9)(15-13) } ; ; T = sqrt{ 1260 } = 35.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 35.5 }{ 8 } = 8.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 35.5 }{ 9 } = 7.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35.5 }{ 13 } = 5.46 ; ;

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**2-9**2-13**2 }{ 2 * 9 * 13 } ) = 37° 21'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-8**2-13**2 }{ 2 * 8 * 13 } ) = 43° 2'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-8**2-9**2 }{ 2 * 9 * 8 } ) = 99° 35'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35.5 }{ 15 } = 2.37 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 37° 21'25" } = 6.59 ; ;




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