10 21 22 triangle

Acute scalene triangle.

Sides: a = 10   b = 21   c = 22

Area: T = 104.0298541757
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 26.76655005768° = 26°45'56″ = 0.4677146111 rad
Angle ∠ B = β = 71.03444250112° = 71°2'4″ = 1.24397845987 rad
Angle ∠ C = γ = 82.22000744121° = 82°12' = 1.43546619439 rad

Height: ha = 20.80657083513
Height: hb = 9.90774801673
Height: hc = 9.45771401597

Median: ma = 20.91765006634
Median: mb = 13.48114687627
Median: mc = 12.22770192606

Inradius: r = 3.92656053493
Circumradius: R = 11.10327222001

Vertex coordinates: A[22; 0] B[0; 0] C[3.25; 9.45771401597]
Centroid: CG[8.41766666667; 3.15223800532]
Coordinates of the circumscribed circle: U[11; 1.50767980129]
Coordinates of the inscribed circle: I[5.5; 3.92656053493]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.2344499423° = 153°14'4″ = 0.4677146111 rad
∠ B' = β' = 108.9665574989° = 108°57'56″ = 1.24397845987 rad
∠ C' = γ' = 97.87999255879° = 97°48' = 1.43546619439 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 = 10 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 10+21+22 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-10)(26.5-21)(26.5-22) } ; ; T = sqrt{ 10821.94 } = 104.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104.03 }{ 10 } = 20.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104.03 }{ 21 } = 9.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104.03 }{ 22 } = 9.46 ; ;

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{ 10**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 26° 45'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-10**2-22**2 }{ 2 * 10 * 22 } ) = 71° 2'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-10**2-21**2 }{ 2 * 21 * 10 } ) = 82° 12' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104.03 }{ 26.5 } = 3.93 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 26° 45'56" } = 11.1 ; ;




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