19 22 27 triangle

Acute scalene triangle.

Sides: a = 19   b = 22   c = 27

Area: T = 206.9788259728
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 44.17985368546° = 44°10'43″ = 0.77110609268 rad
Angle ∠ B = β = 53.79773297664° = 53°47'50″ = 0.93989405332 rad
Angle ∠ C = γ = 82.02441333789° = 82°1'27″ = 1.43215911936 rad

Height: ha = 21.78771852345
Height: hb = 18.81662054298
Height: hc = 15.33217229428

Median: ma = 22.7211135535
Median: mb = 20.5911260282
Median: mc = 15.5

Inradius: r = 6.08875958744
Circumradius: R = 13.63218664758

Vertex coordinates: A[27; 0] B[0; 0] C[11.22222222222; 15.33217229428]
Centroid: CG[12.74107407407; 5.11105743143]
Coordinates of the circumscribed circle: U[13.5; 1.89215030038]
Coordinates of the inscribed circle: I[12; 6.08875958744]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.8211463145° = 135°49'17″ = 0.77110609268 rad
∠ B' = β' = 126.2032670234° = 126°12'10″ = 0.93989405332 rad
∠ C' = γ' = 97.97658666211° = 97°58'33″ = 1.43215911936 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 = 19 ; ; b = 22 ; ; c = 27 ; ;

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

p = a+b+c = 19+22+27 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-19)(34-22)(34-27) } ; ; T = sqrt{ 42840 } = 206.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 206.98 }{ 19 } = 21.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 206.98 }{ 22 } = 18.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 206.98 }{ 27 } = 15.33 ; ;

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{ 19**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 44° 10'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 53° 47'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-19**2-22**2 }{ 2 * 22 * 19 } ) = 82° 1'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 206.98 }{ 34 } = 6.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 44° 10'43" } = 13.63 ; ;




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