12 13 22 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 13   c = 22

Area: T = 65.24113787408
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 27.14443226171° = 27°8'40″ = 0.47437578029 rad
Angle ∠ B = β = 29.62204957505° = 29°37'14″ = 0.51769751769 rad
Angle ∠ C = γ = 123.2355181632° = 123°14'7″ = 2.15108596738 rad

Height: ha = 10.87435631235
Height: hb = 10.03771351909
Height: hc = 5.9311034431

Median: ma = 17.04440605491
Median: mb = 16.48548415218
Median: mc = 5.95881876439

Inradius: r = 2.77662288826
Circumradius: R = 13.15111629055

Vertex coordinates: A[22; 0] B[0; 0] C[10.43218181818; 5.9311034431]
Centroid: CG[10.81106060606; 1.9777011477]
Coordinates of the circumscribed circle: U[11; -7.20878489001]
Coordinates of the inscribed circle: I[10.5; 2.77662288826]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.8565677383° = 152°51'20″ = 0.47437578029 rad
∠ B' = β' = 150.3879504249° = 150°22'46″ = 0.51769751769 rad
∠ C' = γ' = 56.76548183677° = 56°45'53″ = 2.15108596738 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 = 22 ; ;

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

p = a+b+c = 12+13+22 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-12)(23.5-13)(23.5-22) } ; ; T = sqrt{ 4256.44 } = 65.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 65.24 }{ 12 } = 10.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 65.24 }{ 13 } = 10.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 65.24 }{ 22 } = 5.93 ; ;

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**2-22**2 }{ 2 * 13 * 22 } ) = 27° 8'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-22**2 }{ 2 * 12 * 22 } ) = 29° 37'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 123° 14'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 65.24 }{ 23.5 } = 2.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 27° 8'40" } = 13.15 ; ;




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