9 13 15 triangle

Acute scalene triangle.

Sides: a = 9   b = 13   c = 15

Area: T = 58.16551742884
Perimeter: p = 37
Semiperimeter: s = 18.5

Angle ∠ A = α = 36.62443415398° = 36°37'28″ = 0.63992153462 rad
Angle ∠ B = β = 59.50987077655° = 59°30'31″ = 1.03986228841 rad
Angle ∠ C = γ = 83.86769506947° = 83°52'1″ = 1.46437544232 rad

Height: ha = 12.92655942863
Height: hb = 8.94884883521
Height: hc = 7.75553565718

Median: ma = 13.29547358003
Median: mb = 10.52437825899
Median: mc = 8.29215619759

Inradius: r = 3.1444063475
Circumradius: R = 7.54331734774

Vertex coordinates: A[15; 0] B[0; 0] C[4.56766666667; 7.75553565718]
Centroid: CG[6.52222222222; 2.58551188573]
Coordinates of the circumscribed circle: U[7.5; 0.80658946023]
Coordinates of the inscribed circle: I[5.5; 3.1444063475]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.376565846° = 143°22'32″ = 0.63992153462 rad
∠ B' = β' = 120.4911292234° = 120°29'29″ = 1.03986228841 rad
∠ C' = γ' = 96.13330493053° = 96°7'59″ = 1.46437544232 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 = 15 ; ;

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

p = a+b+c = 9+13+15 = 37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-9)(18.5-13)(18.5-15) } ; ; T = sqrt{ 3383.19 } = 58.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.17 }{ 9 } = 12.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.17 }{ 13 } = 8.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.17 }{ 15 } = 7.76 ; ;

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**2-15**2 }{ 2 * 13 * 15 } ) = 36° 37'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-9**2-15**2 }{ 2 * 9 * 15 } ) = 59° 30'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-9**2-13**2 }{ 2 * 13 * 9 } ) = 83° 52'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.17 }{ 18.5 } = 3.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 36° 37'28" } = 7.54 ; ;




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