12 19 22 triangle

Acute scalene triangle.

Sides: a = 12   b = 19   c = 22

Area: T = 113.8799047678
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 33.01661784386° = 33°58″ = 0.57662410202 rad
Angle ∠ B = β = 59.62333765621° = 59°37'24″ = 1.04106242322 rad
Angle ∠ C = γ = 87.36604449993° = 87°21'38″ = 1.52547274012 rad

Height: ha = 18.98798412796
Height: hb = 11.98772681766
Height: hc = 10.3532640698

Median: ma = 19.66596032513
Median: mb = 14.95882753017
Median: mc = 11.46773449412

Inradius: r = 4.29773225539
Circumradius: R = 11.01216832338

Vertex coordinates: A[22; 0] B[0; 0] C[6.06881818182; 10.3532640698]
Centroid: CG[9.35660606061; 3.45108802327]
Coordinates of the circumscribed circle: U[11; 0.5077116991]
Coordinates of the inscribed circle: I[7.5; 4.29773225539]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.9843821561° = 146°59'2″ = 0.57662410202 rad
∠ B' = β' = 120.3776623438° = 120°22'36″ = 1.04106242322 rad
∠ C' = γ' = 92.64395550007° = 92°38'22″ = 1.52547274012 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 = 12 ; ; b = 19 ; ; c = 22 ; ;

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

p = a+b+c = 12+19+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 * (26.5-12)(26.5-19)(26.5-22) } ; ; T = sqrt{ 12968.44 } = 113.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 113.88 }{ 12 } = 18.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 113.88 }{ 19 } = 11.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 113.88 }{ 22 } = 10.35 ; ;

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{ 12**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 33° 58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-12**2-22**2 }{ 2 * 12 * 22 } ) = 59° 37'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-12**2-19**2 }{ 2 * 19 * 12 } ) = 87° 21'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 113.88 }{ 26.5 } = 4.3 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 33° 58" } = 11.01 ; ;




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