13 23 23 triangle

Acute isosceles triangle.

Sides: a = 13   b = 23   c = 23

Area: T = 143.4065674574
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 32.83219234041° = 32°49'55″ = 0.57330251632 rad
Angle ∠ B = β = 73.5844038298° = 73°35'3″ = 1.28442837452 rad
Angle ∠ C = γ = 73.5844038298° = 73°35'3″ = 1.28442837452 rad

Height: ha = 22.06224114729
Height: hb = 12.47700586586
Height: hc = 12.47700586586

Median: ma = 22.06224114729
Median: mb = 14.72224318643
Median: mc = 14.72224318643

Inradius: r = 4.86112093076
Circumradius: R = 11.98987166607

Vertex coordinates: A[23; 0] B[0; 0] C[3.67439130435; 12.47700586586]
Centroid: CG[8.89113043478; 4.15766862195]
Coordinates of the circumscribed circle: U[11.5; 3.3888115578]
Coordinates of the inscribed circle: I[6.5; 4.86112093076]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.1688076596° = 147°10'5″ = 0.57330251632 rad
∠ B' = β' = 106.4165961702° = 106°24'57″ = 1.28442837452 rad
∠ C' = γ' = 106.4165961702° = 106°24'57″ = 1.28442837452 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 = 23 ; ; c = 23 ; ;

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

p = a+b+c = 13+23+23 = 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 * (29.5-13)(29.5-23)(29.5-23) } ; ; T = sqrt{ 20565.19 } = 143.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 143.41 }{ 13 } = 22.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 143.41 }{ 23 } = 12.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 143.41 }{ 23 } = 12.47 ; ;

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**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 32° 49'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 73° 35'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-23**2 }{ 2 * 23 * 13 } ) = 73° 35'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 143.41 }{ 29.5 } = 4.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 32° 49'55" } = 11.99 ; ;




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