22 23 27 triangle

Acute scalene triangle.

Sides: a = 22   b = 23   c = 27

Area: T = 242.8333276138
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 51.45106555599° = 51°27'2″ = 0.89879833418 rad
Angle ∠ B = β = 54.84772970332° = 54°50'50″ = 0.9577265919 rad
Angle ∠ C = γ = 73.70220474069° = 73°42'7″ = 1.28663433927 rad

Height: ha = 22.07657523762
Height: hb = 21.11659370555
Height: hc = 17.98876500843

Median: ma = 22.53988553392
Median: mb = 21.77772817404
Median: mc = 18.00769431054

Inradius: r = 6.74553687816
Circumradius: R = 14.06552057836

Vertex coordinates: A[27; 0] B[0; 0] C[12.66766666667; 17.98876500843]
Centroid: CG[13.22222222222; 5.99658833614]
Coordinates of the circumscribed circle: U[13.5; 3.94771526112]
Coordinates of the inscribed circle: I[13; 6.74553687816]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.549934444° = 128°32'58″ = 0.89879833418 rad
∠ B' = β' = 125.1532702967° = 125°9'10″ = 0.9577265919 rad
∠ C' = γ' = 106.2987952593° = 106°17'53″ = 1.28663433927 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 = 22 ; ; b = 23 ; ; c = 27 ; ;

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

p = a+b+c = 22+23+27 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-22)(36-23)(36-27) } ; ; T = sqrt{ 58968 } = 242.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 242.83 }{ 22 } = 22.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 242.83 }{ 23 } = 21.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 242.83 }{ 27 } = 17.99 ; ;

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{ 22**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 51° 27'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 54° 50'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-22**2-23**2 }{ 2 * 23 * 22 } ) = 73° 42'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 242.83 }{ 36 } = 6.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22 }{ 2 * sin 51° 27'2" } = 14.07 ; ;




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