12 22 22 triangle

Acute isosceles triangle.

Sides: a = 12   b = 22   c = 22

Area: T = 126.9966062931
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 31.65332402637° = 31°39'12″ = 0.55224532615 rad
Angle ∠ B = β = 74.17333798681° = 74°10'24″ = 1.2954569696 rad
Angle ∠ C = γ = 74.17333798681° = 74°10'24″ = 1.2954569696 rad

Height: ha = 21.16660104885
Height: hb = 11.54550966301
Height: hc = 11.54550966301

Median: ma = 21.16660104885
Median: mb = 13.89224439894
Median: mc = 13.89224439894

Inradius: r = 4.53655736761
Circumradius: R = 11.43334253085

Vertex coordinates: A[22; 0] B[0; 0] C[3.27327272727; 11.54550966301]
Centroid: CG[8.42442424242; 3.84883655434]
Coordinates of the circumscribed circle: U[11; 3.11882069023]
Coordinates of the inscribed circle: I[6; 4.53655736761]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.3476759736° = 148°20'48″ = 0.55224532615 rad
∠ B' = β' = 105.8276620132° = 105°49'36″ = 1.2954569696 rad
∠ C' = γ' = 105.8276620132° = 105°49'36″ = 1.2954569696 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 = 22 ; ; c = 22 ; ;

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

p = a+b+c = 12+22+22 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-12)(28-22)(28-22) } ; ; T = sqrt{ 16128 } = 127 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127 }{ 12 } = 21.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127 }{ 22 } = 11.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127 }{ 22 } = 11.55 ; ;

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**2-22**2-22**2 }{ 2 * 22 * 22 } ) = 31° 39'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-12**2-22**2 }{ 2 * 12 * 22 } ) = 74° 10'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-12**2-22**2 }{ 2 * 22 * 12 } ) = 74° 10'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127 }{ 28 } = 4.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 31° 39'12" } = 11.43 ; ;




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