13 13 21 triangle

Obtuse isosceles triangle.

Sides: a = 13   b = 13   c = 21

Area: T = 80.4810976013
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 36.12989274719° = 36°7'44″ = 0.63105687396 rad
Angle ∠ B = β = 36.12989274719° = 36°7'44″ = 0.63105687396 rad
Angle ∠ C = γ = 107.7422145056° = 107°44'32″ = 1.88804551744 rad

Height: ha = 12.38216886174
Height: hb = 12.38216886174
Height: hc = 7.66548548584

Median: ma = 16.21095650774
Median: mb = 16.21095650774
Median: mc = 7.66548548584

Inradius: r = 3.42547223835
Circumradius: R = 11.02443444346

Vertex coordinates: A[21; 0] B[0; 0] C[10.5; 7.66548548584]
Centroid: CG[10.5; 2.55549516195]
Coordinates of the circumscribed circle: U[10.5; -3.35994895762]
Coordinates of the inscribed circle: I[10.5; 3.42547223835]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.8711072528° = 143°52'16″ = 0.63105687396 rad
∠ B' = β' = 143.8711072528° = 143°52'16″ = 0.63105687396 rad
∠ C' = γ' = 72.25878549439° = 72°15'28″ = 1.88804551744 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 = 13 ; ; c = 21 ; ;

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

p = a+b+c = 13+13+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-13)(23.5-13)(23.5-21) } ; ; T = sqrt{ 6477.19 } = 80.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.48 }{ 13 } = 12.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.48 }{ 13 } = 12.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.48 }{ 21 } = 7.66 ; ;

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**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 36° 7'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 36° 7'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-13**2-13**2 }{ 2 * 13 * 13 } ) = 107° 44'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.48 }{ 23.5 } = 3.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 36° 7'44" } = 11.02 ; ;




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