13 20 20 triangle

Acute isosceles triangle.

Sides: a = 13   b = 20   c = 20

Area: T = 122.9432822076
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 37.93111499777° = 37°55'52″ = 0.66220234562 rad
Angle ∠ B = β = 71.03444250112° = 71°2'4″ = 1.24397845987 rad
Angle ∠ C = γ = 71.03444250112° = 71°2'4″ = 1.24397845987 rad

Height: ha = 18.91442803194
Height: hb = 12.29442822076
Height: hc = 12.29442822076

Median: ma = 18.91442803194
Median: mb = 13.58330777072
Median: mc = 13.58330777072

Inradius: r = 4.63993517765
Circumradius: R = 10.57440211429

Vertex coordinates: A[20; 0] B[0; 0] C[4.225; 12.29442822076]
Centroid: CG[8.075; 4.09880940692]
Coordinates of the circumscribed circle: U[10; 3.43765568714]
Coordinates of the inscribed circle: I[6.5; 4.63993517765]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.0698850022° = 142°4'8″ = 0.66220234562 rad
∠ B' = β' = 108.9665574989° = 108°57'56″ = 1.24397845987 rad
∠ C' = γ' = 108.9665574989° = 108°57'56″ = 1.24397845987 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 = 13 ; ; b = 20 ; ; c = 20 ; ;

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

p = a+b+c = 13+20+20 = 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 * (26.5-13)(26.5-20)(26.5-20) } ; ; T = sqrt{ 15114.94 } = 122.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 122.94 }{ 13 } = 18.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 122.94 }{ 20 } = 12.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 122.94 }{ 20 } = 12.29 ; ;

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{ 13**2-20**2-20**2 }{ 2 * 20 * 20 } ) = 37° 55'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-13**2-20**2 }{ 2 * 13 * 20 } ) = 71° 2'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-13**2-20**2 }{ 2 * 20 * 13 } ) = 71° 2'4" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 122.94 }{ 26.5 } = 4.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 37° 55'52" } = 10.57 ; ;




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