8 9 16 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 9   c = 16

Area: T = 22.93333272771
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 18.57333497187° = 18°34'24″ = 0.32441661057 rad
Angle ∠ B = β = 20.99878697385° = 20°59'52″ = 0.36664819628 rad
Angle ∠ C = γ = 140.4298780543° = 140°25'44″ = 2.4510944585 rad

Height: ha = 5.73333318193
Height: hb = 5.09662949505
Height: hc = 2.86766659096

Median: ma = 12.34990890352
Median: mb = 11.82215904175
Median: mc = 2.91554759474

Inradius: r = 1.39898986229
Circumradius: R = 12.55881428512

Vertex coordinates: A[16; 0] B[0; 0] C[7.469875; 2.86766659096]
Centroid: CG[7.82329166667; 0.95655553032]
Coordinates of the circumscribed circle: U[8; -9.68802351145]
Coordinates of the inscribed circle: I[7.5; 1.39898986229]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.4276650281° = 161°25'36″ = 0.32441661057 rad
∠ B' = β' = 159.0022130262° = 159°8″ = 0.36664819628 rad
∠ C' = γ' = 39.57112194572° = 39°34'16″ = 2.4510944585 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 = 8 ; ; b = 9 ; ; c = 16 ; ;

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

p = a+b+c = 8+9+16 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-8)(16.5-9)(16.5-16) } ; ; T = sqrt{ 525.94 } = 22.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.93 }{ 8 } = 5.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.93 }{ 9 } = 5.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.93 }{ 16 } = 2.87 ; ;

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{ 8**2-9**2-16**2 }{ 2 * 9 * 16 } ) = 18° 34'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-8**2-16**2 }{ 2 * 8 * 16 } ) = 20° 59'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-8**2-9**2 }{ 2 * 9 * 8 } ) = 140° 25'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.93 }{ 16.5 } = 1.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 18° 34'24" } = 12.56 ; ;




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