13 21 21 triangle

Acute isosceles triangle.

Sides: a = 13   b = 21   c = 21

Area: T = 129.797671606
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 36.06110707867° = 36°3'40″ = 0.6299384417 rad
Angle ∠ B = β = 71.96994646067° = 71°58'10″ = 1.25661041183 rad
Angle ∠ C = γ = 71.96994646067° = 71°58'10″ = 1.25661041183 rad

Height: ha = 19.96987255477
Height: hb = 12.36215920057
Height: hc = 12.36215920057

Median: ma = 19.96987255477
Median: mb = 13.9555285737
Median: mc = 13.9555285737

Inradius: r = 4.7219880584
Circumradius: R = 11.04222670427

Vertex coordinates: A[21; 0] B[0; 0] C[4.02438095238; 12.36215920057]
Centroid: CG[8.34112698413; 4.12105306686]
Coordinates of the circumscribed circle: U[10.5; 3.41878445608]
Coordinates of the inscribed circle: I[6.5; 4.7219880584]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.9398929213° = 143°56'20″ = 0.6299384417 rad
∠ B' = β' = 108.0310535393° = 108°1'50″ = 1.25661041183 rad
∠ C' = γ' = 108.0310535393° = 108°1'50″ = 1.25661041183 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 = 21 ; ; c = 21 ; ;

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

p = a+b+c = 13+21+21 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-13)(27.5-21)(27.5-21) } ; ; T = sqrt{ 16847.19 } = 129.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.8 }{ 13 } = 19.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.8 }{ 21 } = 12.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.8 }{ 21 } = 12.36 ; ;

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**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 36° 3'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 71° 58'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 71° 58'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.8 }{ 27.5 } = 4.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 36° 3'40" } = 11.04 ; ;




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