20 21 22 triangle

Acute scalene triangle.

Sides: a = 20   b = 21   c = 22

Area: T = 190.0910603397
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 55.37664645208° = 55°22'35″ = 0.9676501634 rad
Angle ∠ B = β = 59.77441988917° = 59°46'27″ = 1.04332565784 rad
Angle ∠ C = γ = 64.84993365875° = 64°50'58″ = 1.13218344412 rad

Height: ha = 19.00990603397
Height: hb = 18.10438669902
Height: hc = 17.28109639452

Median: ma = 19.03994327647
Median: mb = 18.21440056001
Median: mc = 17.30660682999

Inradius: r = 6.03546223301
Circumradius: R = 12.15220998867

Vertex coordinates: A[22; 0] B[0; 0] C[10.06881818182; 17.28109639452]
Centroid: CG[10.68993939394; 5.76603213151]
Coordinates of the circumscribed circle: U[11; 5.16546424518]
Coordinates of the inscribed circle: I[10.5; 6.03546223301]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.6243535479° = 124°37'25″ = 0.9676501634 rad
∠ B' = β' = 120.2265801108° = 120°13'33″ = 1.04332565784 rad
∠ C' = γ' = 115.1510663413° = 115°9'2″ = 1.13218344412 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 = 20 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 20+21+22 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-20)(31.5-21)(31.5-22) } ; ; T = sqrt{ 36134.44 } = 190.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 190.09 }{ 20 } = 19.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 190.09 }{ 21 } = 18.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 190.09 }{ 22 } = 17.28 ; ;

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{ 20**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 55° 22'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-22**2 }{ 2 * 20 * 22 } ) = 59° 46'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 64° 50'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 190.09 }{ 31.5 } = 6.03 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 55° 22'35" } = 12.15 ; ;




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