9 13 20 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 13   c = 20

Area: T = 44.98998886413
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 20.20552235834° = 20°12'19″ = 0.35326476776 rad
Angle ∠ B = β = 29.92664348666° = 29°55'35″ = 0.52223148218 rad
Angle ∠ C = γ = 129.868834155° = 129°52'6″ = 2.26766301542 rad

Height: ha = 9.97877530314
Height: hb = 6.90876751756
Height: hc = 4.49899888641

Median: ma = 16.2565768207
Median: mb = 14.08801278403
Median: mc = 5

Inradius: r = 2.13880899353
Circumradius: R = 13.02989855432

Vertex coordinates: A[20; 0] B[0; 0] C[7.8; 4.49899888641]
Centroid: CG[9.26766666667; 1.49766629547]
Coordinates of the circumscribed circle: U[10; -8.35219138098]
Coordinates of the inscribed circle: I[8; 2.13880899353]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.7954776417° = 159°47'41″ = 0.35326476776 rad
∠ B' = β' = 150.0743565133° = 150°4'25″ = 0.52223148218 rad
∠ C' = γ' = 50.132165845° = 50°7'54″ = 2.26766301542 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 = 20 ; ;

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

p = a+b+c = 9+13+20 = 42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42 }{ 2 } = 21 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21 * (21-9)(21-13)(21-20) } ; ; T = sqrt{ 2016 } = 44.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.9 }{ 9 } = 9.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.9 }{ 13 } = 6.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.9 }{ 20 } = 4.49 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13**2+20**2-9**2 }{ 2 * 13 * 20 } ) = 20° 12'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+20**2-13**2 }{ 2 * 9 * 20 } ) = 29° 55'35" ; ; gamma = 180° - alpha - beta = 180° - 20° 12'19" - 29° 55'35" = 129° 52'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.9 }{ 21 } = 2.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9 }{ 2 * sin 20° 12'19" } = 13.03 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 20**2 - 9**2 } }{ 2 } = 16.256 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 9**2 - 13**2 } }{ 2 } = 14.08 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 9**2 - 20**2 } }{ 2 } = 5 ; ;
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