13 15 21 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 15   c = 21

Area: T = 96.78993976632
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 37.91882026564° = 37°55'6″ = 0.66217974828 rad
Angle ∠ B = β = 45.16600975216° = 45°9'36″ = 0.78881923923 rad
Angle ∠ C = γ = 96.92216998219° = 96°55'18″ = 1.69216027785 rad

Height: ha = 14.89106765636
Height: hb = 12.90552530218
Height: hc = 9.21880378727

Median: ma = 17.0511392905
Median: mb = 15.77218102956
Median: mc = 9.31439680051

Inradius: r = 3.95105876597
Circumradius: R = 10.57770882423

Vertex coordinates: A[21; 0] B[0; 0] C[9.16766666667; 9.21880378727]
Centroid: CG[10.05655555556; 3.07326792909]
Coordinates of the circumscribed circle: U[10.5; -1.27546747369]
Coordinates of the inscribed circle: I[9.5; 3.95105876597]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.0821797344° = 142°4'54″ = 0.66217974828 rad
∠ B' = β' = 134.8439902478° = 134°50'24″ = 0.78881923923 rad
∠ C' = γ' = 83.07883001781° = 83°4'42″ = 1.69216027785 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 = 13 ; ; b = 15 ; ; c = 21 ; ;

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

p = a+b+c = 13+15+21 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-13)(24.5-15)(24.5-21) } ; ; T = sqrt{ 9368.19 } = 96.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 96.79 }{ 13 } = 14.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 96.79 }{ 15 } = 12.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 96.79 }{ 21 } = 9.22 ; ;

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{ 13**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 37° 55'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 45° 9'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 96° 55'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 96.79 }{ 24.5 } = 3.95 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 37° 55'6" } = 10.58 ; ;




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