16 24 27 triangle

Acute scalene triangle.

Sides: a = 16   b = 24   c = 27

Area: T = 190.2655439584
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 35.96113238696° = 35°57'41″ = 0.62876435049 rad
Angle ∠ B = β = 61.74660997912° = 61°44'46″ = 1.07876727416 rad
Angle ∠ C = γ = 82.29325763392° = 82°17'33″ = 1.43662764071 rad

Height: ha = 23.7833179948
Height: hb = 15.85554532986
Height: hc = 14.09437362655

Median: ma = 24.25990189414
Median: mb = 18.66881547026
Median: mc = 15.28988848514

Inradius: r = 5.68795653607
Circumradius: R = 13.62330731428

Vertex coordinates: A[27; 0] B[0; 0] C[7.57440740741; 14.09437362655]
Centroid: CG[11.5254691358; 4.69879120885]
Coordinates of the circumscribed circle: U[13.5; 1.82770527783]
Coordinates of the inscribed circle: I[9.5; 5.68795653607]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.039867613° = 144°2'19″ = 0.62876435049 rad
∠ B' = β' = 118.2543900209° = 118°15'14″ = 1.07876727416 rad
∠ C' = γ' = 97.70774236608° = 97°42'27″ = 1.43662764071 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 = 16 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 16+24+27 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-16)(33.5-24)(33.5-27) } ; ; T = sqrt{ 36200.94 } = 190.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 190.27 }{ 16 } = 23.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 190.27 }{ 24 } = 15.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 190.27 }{ 27 } = 14.09 ; ;

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{ 16**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 35° 57'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-16**2-27**2 }{ 2 * 16 * 27 } ) = 61° 44'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-16**2-24**2 }{ 2 * 24 * 16 } ) = 82° 17'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 190.27 }{ 33.5 } = 5.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 35° 57'41" } = 13.62 ; ;




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