9 21 27 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 21   c = 27

Area: T = 79.07107752586
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 16.1955116739° = 16°11'42″ = 0.28326581098 rad
Angle ∠ B = β = 40.60110607311° = 40°36'4″ = 0.70986221896 rad
Angle ∠ C = γ = 123.204382253° = 123°12'14″ = 2.15503123542 rad

Height: ha = 17.57112833908
Height: hb = 7.53105500246
Height: hc = 5.85770944636

Median: ma = 23.76444692766
Median: mb = 17.16882847134
Median: mc = 8.87441196746

Inradius: r = 2.7744413167
Circumradius: R = 16.13442796479

Vertex coordinates: A[27; 0] B[0; 0] C[6.83333333333; 5.85770944636]
Centroid: CG[11.27877777778; 1.95223648212]
Coordinates of the circumscribed circle: U[13.5; -8.83554388548]
Coordinates of the inscribed circle: I[7.5; 2.7744413167]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.8054883261° = 163°48'18″ = 0.28326581098 rad
∠ B' = β' = 139.3998939269° = 139°23'56″ = 0.70986221896 rad
∠ C' = γ' = 56.796617747° = 56°47'46″ = 2.15503123542 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 = 21 ; ; c = 27 ; ;

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

p = a+b+c = 9+21+27 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-9)(28.5-21)(28.5-27) } ; ; T = sqrt{ 6252.19 } = 79.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 79.07 }{ 9 } = 17.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 79.07 }{ 21 } = 7.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 79.07 }{ 27 } = 5.86 ; ;

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**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 16° 11'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-9**2-27**2 }{ 2 * 9 * 27 } ) = 40° 36'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-9**2-21**2 }{ 2 * 21 * 9 } ) = 123° 12'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 79.07 }{ 28.5 } = 2.77 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 16° 11'42" } = 16.13 ; ;




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