12 22 24 triangle

Acute scalene triangle.

Sides: a = 12   b = 22   c = 24

Area: T = 131.3588288661
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 29.83993144221° = 29°50'22″ = 0.52107942832 rad
Angle ∠ B = β = 65.81326135681° = 65°48'45″ = 1.14986467961 rad
Angle ∠ C = γ = 84.34880720098° = 84°20'53″ = 1.47221515743 rad

Height: ha = 21.89330481102
Height: hb = 11.94216626056
Height: hc = 10.94765240551

Median: ma = 22.22661107709
Median: mb = 15.46596248337
Median: mc = 13.03884048104

Inradius: r = 4.53295961607
Circumradius: R = 12.05986223842

Vertex coordinates: A[24; 0] B[0; 0] C[4.91766666667; 10.94765240551]
Centroid: CG[9.63988888889; 3.64988413517]
Coordinates of the circumscribed circle: U[12; 1.18875915984]
Coordinates of the inscribed circle: I[7; 4.53295961607]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.1610685578° = 150°9'38″ = 0.52107942832 rad
∠ B' = β' = 114.1877386432° = 114°11'15″ = 1.14986467961 rad
∠ C' = γ' = 95.65219279902° = 95°39'7″ = 1.47221515743 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 = 22 ; ; c = 24 ; ;

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

p = a+b+c = 12+22+24 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-12)(29-22)(29-24) } ; ; T = sqrt{ 17255 } = 131.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 131.36 }{ 12 } = 21.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 131.36 }{ 22 } = 11.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 131.36 }{ 24 } = 10.95 ; ;

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**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 29° 50'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 65° 48'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-22**2 }{ 2 * 22 * 12 } ) = 84° 20'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 131.36 }{ 29 } = 4.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 29° 50'22" } = 12.06 ; ;




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