9 13 14 triangle

Acute scalene triangle.

Sides: a = 9   b = 13   c = 14

Area: T = 56.9210997883
Perimeter: p = 36
Semiperimeter: s = 18

Angle ∠ A = α = 38.71992973222° = 38°43'9″ = 0.67657792223 rad
Angle ∠ B = β = 64.62330664748° = 64°37'23″ = 1.12878852827 rad
Angle ∠ C = γ = 76.65876362029° = 76°39'28″ = 1.33879281485 rad

Height: ha = 12.64991106407
Height: hb = 8.75770765974
Height: hc = 8.13215711261

Median: ma = 12.73877392029
Median: mb = 9.81107084352
Median: mc = 8.71877978871

Inradius: r = 3.16222776602
Circumradius: R = 7.19441816769

Vertex coordinates: A[14; 0] B[0; 0] C[3.85771428571; 8.13215711261]
Centroid: CG[5.95223809524; 2.71105237087]
Coordinates of the circumscribed circle: U[7; 1.66601957716]
Coordinates of the inscribed circle: I[5; 3.16222776602]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.2810702678° = 141°16'51″ = 0.67657792223 rad
∠ B' = β' = 115.3776933525° = 115°22'37″ = 1.12878852827 rad
∠ C' = γ' = 103.3422363797° = 103°20'32″ = 1.33879281485 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 = 14 ; ;

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

p = a+b+c = 9+13+14 = 36 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36 }{ 2 } = 18 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18 * (18-9)(18-13)(18-14) } ; ; T = sqrt{ 3240 } = 56.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 56.92 }{ 9 } = 12.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 56.92 }{ 13 } = 8.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 56.92 }{ 14 } = 8.13 ; ;

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+14**2-9**2 }{ 2 * 13 * 14 } ) = 38° 43'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+14**2-13**2 }{ 2 * 9 * 14 } ) = 64° 37'23" ; ; gamma = 180° - alpha - beta = 180° - 38° 43'9" - 64° 37'23" = 76° 39'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 56.92 }{ 18 } = 3.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9 }{ 2 * sin 38° 43'9" } = 7.19 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 14**2 - 9**2 } }{ 2 } = 12.738 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 9**2 - 13**2 } }{ 2 } = 9.811 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 9**2 - 14**2 } }{ 2 } = 8.718 ; ;
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