12 13 13 triangle

Acute isosceles triangle.

Sides: a = 12   b = 13   c = 13

Area: T = 69.1955375568
Perimeter: p = 38
Semiperimeter: s = 19

Angle ∠ A = α = 54.97328525008° = 54°58'22″ = 0.95994572754 rad
Angle ∠ B = β = 62.51435737496° = 62°30'49″ = 1.09110676891 rad
Angle ∠ C = γ = 62.51435737496° = 62°30'49″ = 1.09110676891 rad

Height: ha = 11.53325625947
Height: hb = 10.64554423951
Height: hc = 10.64554423951

Median: ma = 11.53325625947
Median: mb = 10.68987791632
Median: mc = 10.68987791632

Inradius: r = 3.6421861872
Circumradius: R = 7.32770792425

Vertex coordinates: A[13; 0] B[0; 0] C[5.53884615385; 10.64554423951]
Centroid: CG[6.17994871795; 3.54884807984]
Coordinates of the circumscribed circle: U[6.5; 3.38217288811]
Coordinates of the inscribed circle: I[6; 3.6421861872]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.0277147499° = 125°1'38″ = 0.95994572754 rad
∠ B' = β' = 117.486642625° = 117°29'11″ = 1.09110676891 rad
∠ C' = γ' = 117.486642625° = 117°29'11″ = 1.09110676891 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 = 12 ; ; b = 13 ; ; c = 13 ; ;

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

p = a+b+c = 12+13+13 = 38 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-12)(19-13)(19-13) } ; ; T = sqrt{ 4788 } = 69.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.2 }{ 12 } = 11.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.2 }{ 13 } = 10.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.2 }{ 13 } = 10.65 ; ;

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{ 12**2-13**2-13**2 }{ 2 * 13 * 13 } ) = 54° 58'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-13**2 }{ 2 * 12 * 13 } ) = 62° 30'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 62° 30'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.2 }{ 19 } = 3.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 54° 58'22" } = 7.33 ; ;




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