9 22 27 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 22   c = 27

Area: T = 90.11110426086
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 17.662218152° = 17°39'44″ = 0.30882632206 rad
Angle ∠ B = β = 47.87325596059° = 47°52'21″ = 0.83655337865 rad
Angle ∠ C = γ = 114.4655258874° = 114°27'55″ = 1.99877956465 rad

Height: ha = 20.02546761352
Height: hb = 8.19219129644
Height: hc = 6.67548920451

Median: ma = 24.21326000256
Median: mb = 16.85222995464
Median: mc = 10.01224921973

Inradius: r = 3.10772773313
Circumradius: R = 14.83217005476

Vertex coordinates: A[27; 0] B[0; 0] C[6.0377037037; 6.67548920451]
Centroid: CG[11.0122345679; 2.2254964015]
Coordinates of the circumscribed circle: U[13.5; -6.14224214389]
Coordinates of the inscribed circle: I[7; 3.10772773313]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.338781848° = 162°20'16″ = 0.30882632206 rad
∠ B' = β' = 132.1277440394° = 132°7'39″ = 0.83655337865 rad
∠ C' = γ' = 65.53547411259° = 65°32'5″ = 1.99877956465 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 = 9 ; ; b = 22 ; ; c = 27 ; ;

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

p = a+b+c = 9+22+27 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-9)(29-22)(29-27) } ; ; T = sqrt{ 8120 } = 90.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90.11 }{ 9 } = 20.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90.11 }{ 22 } = 8.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90.11 }{ 27 } = 6.67 ; ;

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{ 9**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 17° 39'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-9**2-27**2 }{ 2 * 9 * 27 } ) = 47° 52'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-9**2-22**2 }{ 2 * 22 * 9 } ) = 114° 27'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90.11 }{ 29 } = 3.11 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 17° 39'44" } = 14.83 ; ;




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