6 10 11 triangle

Acute scalene triangle.

Sides: a = 6   b = 10   c = 11

Area: T = 29.76547022495
Perimeter: p = 27
Semiperimeter: s = 13.5

Angle ∠ A = α = 32.76437577589° = 32°45'50″ = 0.57218354482 rad
Angle ∠ B = β = 64.41769980226° = 64°25'1″ = 1.12442887097 rad
Angle ∠ C = γ = 82.81992442185° = 82°49'9″ = 1.44554684956 rad

Height: ha = 9.92215674165
Height: hb = 5.95329404499
Height: hc = 5.41217640454

Median: ma = 10.07547208398
Median: mb = 7.31443694192
Median: mc = 6.14441028637

Inradius: r = 2.20547927592
Circumradius: R = 5.54334789375

Vertex coordinates: A[11; 0] B[0; 0] C[2.59109090909; 5.41217640454]
Centroid: CG[4.53303030303; 1.80439213485]
Coordinates of the circumscribed circle: U[5.5; 0.69329348672]
Coordinates of the inscribed circle: I[3.5; 2.20547927592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.2366242241° = 147°14'10″ = 0.57218354482 rad
∠ B' = β' = 115.5833001977° = 115°34'59″ = 1.12442887097 rad
∠ C' = γ' = 97.18107557815° = 97°10'51″ = 1.44554684956 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 = 6 ; ; b = 10 ; ; c = 11 ; ;

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

p = a+b+c = 6+10+11 = 27 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27 }{ 2 } = 13.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.5 * (13.5-6)(13.5-10)(13.5-11) } ; ; T = sqrt{ 885.94 } = 29.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.76 }{ 6 } = 9.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.76 }{ 10 } = 5.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.76 }{ 11 } = 5.41 ; ;

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{ 6**2-10**2-11**2 }{ 2 * 10 * 11 } ) = 32° 45'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-6**2-11**2 }{ 2 * 6 * 11 } ) = 64° 25'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11**2-6**2-10**2 }{ 2 * 10 * 6 } ) = 82° 49'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.76 }{ 13.5 } = 2.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 32° 45'50" } = 5.54 ; ;




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