11 20 23 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 20   c = 23

Area: T = 109.9821816679
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 28.56767204538° = 28°34' = 0.49985833284 rad
Angle ∠ B = β = 60.39114806958° = 60°23'29″ = 1.05440301783 rad
Angle ∠ C = γ = 91.04217988505° = 91°2'30″ = 1.58989791469 rad

Height: ha = 19.99766939416
Height: hb = 10.99881816679
Height: hc = 9.5643636233

Median: ma = 20.83986659842
Median: mb = 15
Median: mc = 11.32547516529

Inradius: r = 4.07334006177
Circumradius: R = 11.50219012979

Vertex coordinates: A[23; 0] B[0; 0] C[5.43547826087; 9.5643636233]
Centroid: CG[9.47882608696; 3.18878787443]
Coordinates of the circumscribed circle: U[11.5; -0.20991254781]
Coordinates of the inscribed circle: I[7; 4.07334006177]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.4333279546° = 151°26' = 0.49985833284 rad
∠ B' = β' = 119.6098519304° = 119°36'31″ = 1.05440301783 rad
∠ C' = γ' = 88.95882011495° = 88°57'30″ = 1.58989791469 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 = 20 ; ; c = 23 ; ;

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

p = a+b+c = 11+20+23 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-11)(27-20)(27-23) } ; ; T = sqrt{ 12096 } = 109.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.98 }{ 11 } = 20 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.98 }{ 20 } = 11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.98 }{ 23 } = 9.56 ; ;

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**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 28° 34' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-11**2-23**2 }{ 2 * 11 * 23 } ) = 60° 23'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-11**2-20**2 }{ 2 * 20 * 11 } ) = 91° 2'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.98 }{ 27 } = 4.07 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 28° 34' } = 11.5 ; ;




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