9 13 21 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 13   c = 21

Area: T = 33.79662645865
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 14.33550459687° = 14°20'6″ = 0.25501937506 rad
Angle ∠ B = β = 20.95548664178° = 20°57'18″ = 0.36657314133 rad
Angle ∠ C = γ = 144.7110087613° = 144°42'36″ = 2.52656674897 rad

Height: ha = 7.51102810192
Height: hb = 5.1999425321
Height: hc = 3.21986918654

Median: ma = 16.87545370307
Median: mb = 14.79901994577
Median: mc = 3.84105728739

Inradius: r = 1.57219192831
Circumradius: R = 18.17550855462

Vertex coordinates: A[21; 0] B[0; 0] C[8.40547619048; 3.21986918654]
Centroid: CG[9.80215873016; 1.07328972885]
Coordinates of the circumscribed circle: U[10.5; -14.83552193988]
Coordinates of the inscribed circle: I[8.5; 1.57219192831]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.6654954031° = 165°39'54″ = 0.25501937506 rad
∠ B' = β' = 159.0455133582° = 159°2'42″ = 0.36657314133 rad
∠ C' = γ' = 35.29899123865° = 35°17'24″ = 2.52656674897 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 = 13 ; ; c = 21 ; ;

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

p = a+b+c = 9+13+21 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-9)(21.5-13)(21.5-21) } ; ; T = sqrt{ 1142.19 } = 33.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 33.8 }{ 9 } = 7.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 33.8 }{ 13 } = 5.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 33.8 }{ 21 } = 3.22 ; ;

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{ 9**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 14° 20'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-9**2-21**2 }{ 2 * 9 * 21 } ) = 20° 57'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-9**2-13**2 }{ 2 * 13 * 9 } ) = 144° 42'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 33.8 }{ 21.5 } = 1.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 14° 20'6" } = 18.18 ; ;




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