10 20 22 triangle

Acute scalene triangle.

Sides: a = 10   b = 20   c = 22

Area: T = 99.92199679744
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 27.01222939589° = 27°44″ = 0.47114534681 rad
Angle ∠ B = β = 65.28801488171° = 65°16'49″ = 1.13993535331 rad
Angle ∠ C = γ = 87.7087557224° = 87°42'27″ = 1.53107856524 rad

Height: ha = 19.98439935949
Height: hb = 9.99219967974
Height: hc = 9.08436334522

Median: ma = 20.42105778567
Median: mb = 13.85664064606
Median: mc = 11.35878166916

Inradius: r = 3.84330756913
Circumradius: R = 11.00988105741

Vertex coordinates: A[22; 0] B[0; 0] C[4.18218181818; 9.08436334522]
Centroid: CG[8.72772727273; 3.02878778174]
Coordinates of the circumscribed circle: U[11; 0.4440352423]
Coordinates of the inscribed circle: I[6; 3.84330756913]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.9887706041° = 152°59'16″ = 0.47114534681 rad
∠ B' = β' = 114.7219851183° = 114°43'11″ = 1.13993535331 rad
∠ C' = γ' = 92.2922442776° = 92°17'33″ = 1.53107856524 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 = 20 ; ; c = 22 ; ;

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

p = a+b+c = 10+20+22 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-10)(26-20)(26-22) } ; ; T = sqrt{ 9984 } = 99.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 99.92 }{ 10 } = 19.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 99.92 }{ 20 } = 9.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 99.92 }{ 22 } = 9.08 ; ;

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**2-20**2-22**2 }{ 2 * 20 * 22 } ) = 27° 44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-10**2-22**2 }{ 2 * 10 * 22 } ) = 65° 16'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-10**2-20**2 }{ 2 * 20 * 10 } ) = 87° 42'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 99.92 }{ 26 } = 3.84 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 27° 44" } = 11.01 ; ;




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