11 21 22 triangle

Acute scalene triangle.

Sides: a = 11   b = 21   c = 22

Area: T = 113.8421995766
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 29.52662652473° = 29°31'35″ = 0.51553305444 rad
Angle ∠ B = β = 70.19436024552° = 70°11'37″ = 1.22551094767 rad
Angle ∠ C = γ = 80.28801322976° = 80°16'48″ = 1.40111526325 rad

Height: ha = 20.69985446847
Height: hb = 10.84220948349
Height: hc = 10.34992723424

Median: ma = 20.79106228863
Median: mb = 13.86554246239
Median: mc = 12.64991106407

Inradius: r = 4.21663702136
Circumradius: R = 11.1660204909

Vertex coordinates: A[22; 0] B[0; 0] C[3.72772727273; 10.34992723424]
Centroid: CG[8.57657575758; 3.45497574475]
Coordinates of the circumscribed circle: U[11; 1.88441904392]
Coordinates of the inscribed circle: I[6; 4.21663702136]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.4743734753° = 150°28'25″ = 0.51553305444 rad
∠ B' = β' = 109.8066397545° = 109°48'23″ = 1.22551094767 rad
∠ C' = γ' = 99.72198677024° = 99°43'12″ = 1.40111526325 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 = 22 ; ;

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

p = a+b+c = 11+21+22 = 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)(27-21)(27-22) } ; ; T = sqrt{ 12960 } = 113.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 113.84 }{ 11 } = 20.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 113.84 }{ 21 } = 10.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 113.84 }{ 22 } = 10.35 ; ;

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**2-22**2 }{ 2 * 21 * 22 } ) = 29° 31'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-11**2-22**2 }{ 2 * 11 * 22 } ) = 70° 11'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-11**2-21**2 }{ 2 * 21 * 11 } ) = 80° 16'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 113.84 }{ 27 } = 4.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 29° 31'35" } = 11.16 ; ;




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