13 15 22 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 15   c = 22

Area: T = 94.86883298051
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 35.09768012276° = 35°5'48″ = 0.61325547383 rad
Angle ∠ B = β = 41.56108349753° = 41°33'39″ = 0.72553734102 rad
Angle ∠ C = γ = 103.3422363797° = 103°20'32″ = 1.80436645051 rad

Height: ha = 14.59551276623
Height: hb = 12.64991106407
Height: hc = 8.62443936186

Median: ma = 17.67105970471
Median: mb = 16.43992822228
Median: mc = 8.71877978871

Inradius: r = 3.79547331922
Circumradius: R = 11.30551426351

Vertex coordinates: A[22; 0] B[0; 0] C[9.72772727273; 8.62443936186]
Centroid: CG[10.57657575758; 2.87547978729]
Coordinates of the circumscribed circle: U[11; -2.60988790696]
Coordinates of the inscribed circle: I[10; 3.79547331922]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.9033198772° = 144°54'12″ = 0.61325547383 rad
∠ B' = β' = 138.4399165025° = 138°26'21″ = 0.72553734102 rad
∠ C' = γ' = 76.65876362029° = 76°39'28″ = 1.80436645051 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 = 13 ; ; b = 15 ; ; c = 22 ; ;

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

p = a+b+c = 13+15+22 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-13)(25-15)(25-22) } ; ; T = sqrt{ 9000 } = 94.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 94.87 }{ 13 } = 14.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 94.87 }{ 15 } = 12.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 94.87 }{ 22 } = 8.62 ; ;

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{ 13**2-15**2-22**2 }{ 2 * 15 * 22 } ) = 35° 5'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-22**2 }{ 2 * 13 * 22 } ) = 41° 33'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 103° 20'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 94.87 }{ 25 } = 3.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 35° 5'48" } = 11.31 ; ;




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