11 13 15 triangle

Acute scalene triangle.

Sides: a = 11   b = 13   c = 15

Area: T = 69.62989271783
Perimeter: p = 39
Semiperimeter: s = 19.5

Angle ∠ A = α = 45.57329959992° = 45°34'23″ = 0.79553988302 rad
Angle ∠ B = β = 57.56435627878° = 57°33'49″ = 1.00546736998 rad
Angle ∠ C = γ = 76.86334412131° = 76°51'48″ = 1.34215201236 rad

Height: ha = 12.66598049415
Height: hb = 10.71221426428
Height: hc = 9.28438569571

Median: ma = 12.91331715701
Median: mb = 11.4354596626
Median: mc = 9.42107218407

Inradius: r = 3.57107142143
Circumradius: R = 7.70215404622

Vertex coordinates: A[15; 0] B[0; 0] C[5.9; 9.28438569571]
Centroid: CG[6.96766666667; 3.09546189857]
Coordinates of the circumscribed circle: U[7.5; 1.7550350105]
Coordinates of the inscribed circle: I[6.5; 3.57107142143]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.4277004001° = 134°25'37″ = 0.79553988302 rad
∠ B' = β' = 122.4366437212° = 122°26'11″ = 1.00546736998 rad
∠ C' = γ' = 103.1376558787° = 103°8'12″ = 1.34215201236 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 = 11 ; ; b = 13 ; ; c = 15 ; ;

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

p = a+b+c = 11+13+15 = 39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39 }{ 2 } = 19.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.5 * (19.5-11)(19.5-13)(19.5-15) } ; ; T = sqrt{ 4848.19 } = 69.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.63 }{ 11 } = 12.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.63 }{ 13 } = 10.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.63 }{ 15 } = 9.28 ; ;

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{ 11**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 45° 34'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-11**2-15**2 }{ 2 * 11 * 15 } ) = 57° 33'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-11**2-13**2 }{ 2 * 13 * 11 } ) = 76° 51'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.63 }{ 19.5 } = 3.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 45° 34'23" } = 7.7 ; ;




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