13 21 22 triangle

Acute scalene triangle.

Sides: a = 13   b = 21   c = 22

Area: T = 132.8165661727
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 35.09768012276° = 35°5'48″ = 0.61325547383 rad
Angle ∠ B = β = 68.24655625695° = 68°14'44″ = 1.19111097667 rad
Angle ∠ C = γ = 76.65876362029° = 76°39'28″ = 1.33879281485 rad

Height: ha = 20.43331787272
Height: hb = 12.64991106407
Height: hc = 12.07441510661

Median: ma = 20.5
Median: mb = 14.70554411699
Median: mc = 13.56546599663

Inradius: r = 4.74334164903
Circumradius: R = 11.30551426351

Vertex coordinates: A[22; 0] B[0; 0] C[4.81881818182; 12.07441510661]
Centroid: CG[8.93993939394; 4.0254717022]
Coordinates of the circumscribed circle: U[11; 2.60988790696]
Coordinates of the inscribed circle: I[7; 4.74334164903]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.9033198772° = 144°54'12″ = 0.61325547383 rad
∠ B' = β' = 111.754443743° = 111°45'16″ = 1.19111097667 rad
∠ C' = γ' = 103.3422363797° = 103°20'32″ = 1.33879281485 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 = 13 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 13+21+22 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-13)(28-21)(28-22) } ; ; T = sqrt{ 17640 } = 132.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 132.82 }{ 13 } = 20.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 132.82 }{ 21 } = 12.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 132.82 }{ 22 } = 12.07 ; ;

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{ 13**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 35° 5'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-22**2 }{ 2 * 13 * 22 } ) = 68° 14'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 76° 39'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 132.82 }{ 28 } = 4.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 35° 5'48" } = 11.31 ; ;




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