13 24 24 triangle

Acute isosceles triangle.

Sides: a = 13   b = 24   c = 24

Area: T = 150.1769695678
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 31.4287722096° = 31°25'40″ = 0.5498517227 rad
Angle ∠ B = β = 74.2866138952° = 74°17'10″ = 1.29765377133 rad
Angle ∠ C = γ = 74.2866138952° = 74°17'10″ = 1.29765377133 rad

Height: ha = 23.10330301043
Height: hb = 12.51441413065
Height: hc = 12.51441413065

Median: ma = 23.10330301043
Median: mb = 15.11662164578
Median: mc = 15.11662164578

Inradius: r = 4.92435965796
Circumradius: R = 12.46658972741

Vertex coordinates: A[24; 0] B[0; 0] C[3.52108333333; 12.51441413065]
Centroid: CG[9.17436111111; 4.17113804355]
Coordinates of the circumscribed circle: U[12; 3.37661805117]
Coordinates of the inscribed circle: I[6.5; 4.92435965796]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.5722277904° = 148°34'20″ = 0.5498517227 rad
∠ B' = β' = 105.7143861048° = 105°42'50″ = 1.29765377133 rad
∠ C' = γ' = 105.7143861048° = 105°42'50″ = 1.29765377133 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 = 24 ; ; c = 24 ; ;

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

p = a+b+c = 13+24+24 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-13)(30.5-24)(30.5-24) } ; ; T = sqrt{ 22550.94 } = 150.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 150.17 }{ 13 } = 23.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 150.17 }{ 24 } = 12.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 150.17 }{ 24 } = 12.51 ; ;

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**2-24**2-24**2 }{ 2 * 24 * 24 } ) = 31° 25'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 74° 17'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-13**2-24**2 }{ 2 * 24 * 13 } ) = 74° 17'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 150.17 }{ 30.5 } = 4.92 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 31° 25'40" } = 12.47 ; ;




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