9 12 12 triangle

Acute isosceles triangle.

Sides: a = 9   b = 12   c = 12

Area: T = 50.05993397879
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 44.04986256741° = 44°2'55″ = 0.7698793549 rad
Angle ∠ B = β = 67.9765687163° = 67°58'32″ = 1.18663995523 rad
Angle ∠ C = γ = 67.9765687163° = 67°58'32″ = 1.18663995523 rad

Height: ha = 11.12442977306
Height: hb = 8.3433223298
Height: hc = 8.3433223298

Median: ma = 11.12442977306
Median: mb = 8.74664278423
Median: mc = 8.74664278423

Inradius: r = 3.03438993811
Circumradius: R = 6.47223186796

Vertex coordinates: A[12; 0] B[0; 0] C[3.375; 8.3433223298]
Centroid: CG[5.125; 2.78110744327]
Coordinates of the circumscribed circle: U[6; 2.42771195049]
Coordinates of the inscribed circle: I[4.5; 3.03438993811]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.9511374326° = 135°57'5″ = 0.7698793549 rad
∠ B' = β' = 112.0244312837° = 112°1'28″ = 1.18663995523 rad
∠ C' = γ' = 112.0244312837° = 112°1'28″ = 1.18663995523 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 = 9 ; ; b = 12 ; ; c = 12 ; ;

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

p = a+b+c = 9+12+12 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-9)(16.5-12)(16.5-12) } ; ; T = sqrt{ 2505.94 } = 50.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50.06 }{ 9 } = 11.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50.06 }{ 12 } = 8.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50.06 }{ 12 } = 8.34 ; ;

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{ 9**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 44° 2'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 67° 58'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-9**2-12**2 }{ 2 * 12 * 9 } ) = 67° 58'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50.06 }{ 16.5 } = 3.03 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 44° 2'55" } = 6.47 ; ;




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