11 20 22 triangle

Acute scalene triangle.

Sides: a = 11   b = 20   c = 22

Area: T = 109.6110389562
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 29.88329034608° = 29°52'58″ = 0.52215550554 rad
Angle ∠ B = β = 64.94108471732° = 64°56'27″ = 1.13334316022 rad
Angle ∠ C = γ = 85.1766249366° = 85°10'34″ = 1.48766059959 rad

Height: ha = 19.92991617386
Height: hb = 10.96110389562
Height: hc = 9.96545808693

Median: ma = 20.29216238877
Median: mb = 14.23302494708
Median: mc = 11.8111011811

Inradius: r = 4.13662411156
Circumradius: R = 11.03990995309

Vertex coordinates: A[22; 0] B[0; 0] C[4.65990909091; 9.96545808693]
Centroid: CG[8.88663636364; 3.32215269564]
Coordinates of the circumscribed circle: U[11; 0.92882879151]
Coordinates of the inscribed circle: I[6.5; 4.13662411156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.1177096539° = 150°7'2″ = 0.52215550554 rad
∠ B' = β' = 115.0599152827° = 115°3'33″ = 1.13334316022 rad
∠ C' = γ' = 94.8243750634° = 94°49'25″ = 1.48766059959 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 = 20 ; ; c = 22 ; ;

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

p = a+b+c = 11+20+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 * (26.5-11)(26.5-20)(26.5-22) } ; ; T = sqrt{ 12014.44 } = 109.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.61 }{ 11 } = 19.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.61 }{ 20 } = 10.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.61 }{ 22 } = 9.96 ; ;

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**2-20**2-22**2 }{ 2 * 20 * 22 } ) = 29° 52'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-11**2-22**2 }{ 2 * 11 * 22 } ) = 64° 56'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-11**2-20**2 }{ 2 * 20 * 11 } ) = 85° 10'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.61 }{ 26.5 } = 4.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 29° 52'58" } = 11.04 ; ;




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