16 22 27 triangle

Acute scalene triangle.

Sides: a = 16   b = 22   c = 27

Area: T = 175.9798514313
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 36.33660575146° = 36°20'10″ = 0.63441838408 rad
Angle ∠ B = β = 54.5599225472° = 54°33'33″ = 0.95222381218 rad
Angle ∠ C = γ = 89.10547170134° = 89°6'17″ = 1.55551706909 rad

Height: ha = 21.99773142892
Height: hb = 15.99880467558
Height: hc = 13.03554455047

Median: ma = 23.29216293977
Median: mb = 19.27443352674
Median: mc = 13.7022189606

Inradius: r = 5.41547235173
Circumradius: R = 13.5021648251

Vertex coordinates: A[27; 0] B[0; 0] C[9.27877777778; 13.03554455047]
Centroid: CG[12.09325925926; 4.34551485016]
Coordinates of the circumscribed circle: U[13.5; 0.21109632539]
Coordinates of the inscribed circle: I[10.5; 5.41547235173]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.6643942485° = 143°39'50″ = 0.63441838408 rad
∠ B' = β' = 125.4410774528° = 125°26'27″ = 0.95222381218 rad
∠ C' = γ' = 90.89552829866° = 90°53'43″ = 1.55551706909 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 = 22 ; ; c = 27 ; ;

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

p = a+b+c = 16+22+27 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-16)(32.5-22)(32.5-27) } ; ; T = sqrt{ 30968.44 } = 175.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 175.98 }{ 16 } = 22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 175.98 }{ 22 } = 16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 175.98 }{ 27 } = 13.04 ; ;

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**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 36° 20'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-16**2-27**2 }{ 2 * 16 * 27 } ) = 54° 33'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-16**2-22**2 }{ 2 * 22 * 16 } ) = 89° 6'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 175.98 }{ 32.5 } = 5.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 36° 20'10" } = 13.5 ; ;




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