12 13 15 triangle

Acute scalene triangle.

Sides: a = 12   b = 13   c = 15

Area: T = 74.83331477355
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 50.132165845° = 50°7'54″ = 0.87549624994 rad
Angle ∠ B = β = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ C = γ = 73.61773301459° = 73°37'2″ = 1.28548647976 rad

Height: ha = 12.47221912892
Height: hb = 11.51327919593
Height: hc = 9.97877530314

Median: ma = 12.68985775404
Median: mb = 11.92768604419
Median: mc = 10.01224921973

Inradius: r = 3.74216573868
Circumradius: R = 7.81773913259

Vertex coordinates: A[15; 0] B[0; 0] C[6.66766666667; 9.97877530314]
Centroid: CG[7.22222222222; 3.32659176771]
Coordinates of the circumscribed circle: U[7.5; 2.20549052458]
Coordinates of the inscribed circle: I[7; 3.74216573868]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.868834155° = 129°52'6″ = 0.87549624994 rad
∠ B' = β' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ C' = γ' = 106.3832669854° = 106°22'58″ = 1.28548647976 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 = 12 ; ; b = 13 ; ; c = 15 ; ;

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

p = a+b+c = 12+13+15 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-12)(20-13)(20-15) } ; ; T = sqrt{ 5600 } = 74.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74.83 }{ 12 } = 12.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74.83 }{ 13 } = 11.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74.83 }{ 15 } = 9.98 ; ;

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{ 12**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 50° 7'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 56° 15'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 73° 37'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74.83 }{ 20 } = 3.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 50° 7'54" } = 7.82 ; ;




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