11 22 24 triangle

Acute scalene triangle.

Sides: a = 11   b = 22   c = 24

Area: T = 120.7832604294
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 27.227653999° = 27°13'36″ = 0.47551927668 rad
Angle ∠ B = β = 66.20985290545° = 66°12'31″ = 1.15655568249 rad
Angle ∠ C = γ = 86.56549309555° = 86°33'54″ = 1.51108430619 rad

Height: ha = 21.9660473508
Height: hb = 10.9880236754
Height: hc = 10.06552170245

Median: ma = 22.35550889061
Median: mb = 15.0833103129
Median: mc = 12.5989678312

Inradius: r = 4.23879861156
Circumradius: R = 12.02215987103

Vertex coordinates: A[24; 0] B[0; 0] C[4.43875; 10.06552170245]
Centroid: CG[9.47991666667; 3.35550723415]
Coordinates of the circumscribed circle: U[12; 0.72203024021]
Coordinates of the inscribed circle: I[6.5; 4.23879861156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.773346001° = 152°46'24″ = 0.47551927668 rad
∠ B' = β' = 113.7911470945° = 113°47'29″ = 1.15655568249 rad
∠ C' = γ' = 93.43550690445° = 93°26'6″ = 1.51108430619 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 = 11 ; ; b = 22 ; ; c = 24 ; ;

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

p = a+b+c = 11+22+24 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-11)(28.5-22)(28.5-24) } ; ; T = sqrt{ 14588.44 } = 120.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 120.78 }{ 11 } = 21.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 120.78 }{ 22 } = 10.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 120.78 }{ 24 } = 10.07 ; ;

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{ 11**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 27° 13'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-11**2-24**2 }{ 2 * 11 * 24 } ) = 66° 12'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-11**2-22**2 }{ 2 * 22 * 11 } ) = 86° 33'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 120.78 }{ 28.5 } = 4.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 27° 13'36" } = 12.02 ; ;




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