15 22 22 triangle

Acute isosceles triangle.

Sides: a = 15   b = 22   c = 22

Area: T = 155.11658841
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 39.86545409444° = 39°51'52″ = 0.69657674943 rad
Angle ∠ B = β = 70.06877295278° = 70°4'4″ = 1.22329125797 rad
Angle ∠ C = γ = 70.06877295278° = 70°4'4″ = 1.22329125797 rad

Height: ha = 20.68221178799
Height: hb = 14.10114440091
Height: hc = 14.10114440091

Median: ma = 20.68221178799
Median: mb = 15.28107067899
Median: mc = 15.28107067899

Inradius: r = 5.25881655627
Circumradius: R = 11.70109293441

Vertex coordinates: A[22; 0] B[0; 0] C[5.11436363636; 14.10114440091]
Centroid: CG[9.03878787879; 4.77004813364]
Coordinates of the circumscribed circle: U[11; 3.98989531855]
Coordinates of the inscribed circle: I[7.5; 5.25881655627]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.1355459056° = 140°8'8″ = 0.69657674943 rad
∠ B' = β' = 109.9322270472° = 109°55'56″ = 1.22329125797 rad
∠ C' = γ' = 109.9322270472° = 109°55'56″ = 1.22329125797 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 = 15 ; ; b = 22 ; ; c = 22 ; ;

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

p = a+b+c = 15+22+22 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-15)(29.5-22)(29.5-22) } ; ; T = sqrt{ 24060.94 } = 155.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 155.12 }{ 15 } = 20.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 155.12 }{ 22 } = 14.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 155.12 }{ 22 } = 14.1 ; ;

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{ 15**2-22**2-22**2 }{ 2 * 22 * 22 } ) = 39° 51'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-15**2-22**2 }{ 2 * 15 * 22 } ) = 70° 4'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-15**2-22**2 }{ 2 * 22 * 15 } ) = 70° 4'4" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 155.12 }{ 29.5 } = 5.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 39° 51'52" } = 11.7 ; ;




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