9 11 13 triangle

Acute scalene triangle.

Sides: a = 9   b = 11   c = 13

Area: T = 48.80876582106
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 43.04990798002° = 43°2'57″ = 0.75113481825 rad
Angle ∠ B = β = 56.54549884266° = 56°32'42″ = 0.98768962235 rad
Angle ∠ C = γ = 80.40659317731° = 80°24'21″ = 1.40333482476 rad

Height: ha = 10.8466146269
Height: hb = 8.87441196746
Height: hc = 7.50988704939

Median: ma = 11.16991539518
Median: mb = 9.7343961167
Median: mc = 7.66548548584

Inradius: r = 2.95880398915
Circumradius: R = 6.59222031869

Vertex coordinates: A[13; 0] B[0; 0] C[4.96215384615; 7.50988704939]
Centroid: CG[5.98771794872; 2.50329568313]
Coordinates of the circumscribed circle: U[6.5; 1.09987005311]
Coordinates of the inscribed circle: I[5.5; 2.95880398915]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.95109202° = 136°57'3″ = 0.75113481825 rad
∠ B' = β' = 123.4555011573° = 123°27'18″ = 0.98768962235 rad
∠ C' = γ' = 99.59440682269° = 99°35'39″ = 1.40333482476 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 = 9 ; ; b = 11 ; ; c = 13 ; ;

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

p = a+b+c = 9+11+13 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-9)(16.5-11)(16.5-13) } ; ; T = sqrt{ 2382.19 } = 48.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 48.81 }{ 9 } = 10.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 48.81 }{ 11 } = 8.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 48.81 }{ 13 } = 7.51 ; ;

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{ 9**2-11**2-13**2 }{ 2 * 11 * 13 } ) = 43° 2'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-9**2-13**2 }{ 2 * 9 * 13 } ) = 56° 32'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-9**2-11**2 }{ 2 * 11 * 9 } ) = 80° 24'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 48.81 }{ 16.5 } = 2.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 43° 2'57" } = 6.59 ; ;




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