11 21 27 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 21   c = 27

Area: T = 107.6990238648
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 22.32549954181° = 22°19'30″ = 0.39896446755 rad
Angle ∠ B = β = 46.4844388962° = 46°29'4″ = 0.81113056382 rad
Angle ∠ C = γ = 111.191061562° = 111°11'26″ = 1.94106423399 rad

Height: ha = 19.58800433905
Height: hb = 10.25662132045
Height: hc = 7.97770547146

Median: ma = 23.55331314266
Median: mb = 17.74111949992
Median: mc = 9.93773034572

Inradius: r = 3.65105165643
Circumradius: R = 14.47990281792

Vertex coordinates: A[27; 0] B[0; 0] C[7.57440740741; 7.97770547146]
Centroid: CG[11.5254691358; 2.65990182382]
Coordinates of the circumscribed circle: U[13.5; -5.23437612682]
Coordinates of the inscribed circle: I[8.5; 3.65105165643]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.6755004582° = 157°40'30″ = 0.39896446755 rad
∠ B' = β' = 133.5165611038° = 133°30'56″ = 0.81113056382 rad
∠ C' = γ' = 68.80993843802° = 68°48'34″ = 1.94106423399 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 = 11 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 11+21+27 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-11)(29.5-21)(29.5-27) } ; ; T = sqrt{ 11597.19 } = 107.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 107.69 }{ 11 } = 19.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 107.69 }{ 21 } = 10.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 107.69 }{ 27 } = 7.98 ; ;

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{ 11**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 22° 19'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-11**2-27**2 }{ 2 * 11 * 27 } ) = 46° 29'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-11**2-21**2 }{ 2 * 21 * 11 } ) = 111° 11'26" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 107.69 }{ 29.5 } = 3.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 22° 19'30" } = 14.48 ; ;




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