13 30 30 triangle

Acute isosceles triangle.

Sides: a = 13   b = 30   c = 30

Area: T = 190.3687900393
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 25.02766507258° = 25°1'36″ = 0.43767974559 rad
Angle ∠ B = β = 77.48766746371° = 77°29'12″ = 1.35223975988 rad
Angle ∠ C = γ = 77.48766746371° = 77°29'12″ = 1.35223975988 rad

Height: ha = 29.28773692912
Height: hb = 12.69111933595
Height: hc = 12.69111933595

Median: ma = 29.28773692912
Median: mb = 17.59326120858
Median: mc = 17.59326120858

Inradius: r = 5.21655589149
Circumradius: R = 15.36549853466

Vertex coordinates: A[30; 0] B[0; 0] C[2.81766666667; 12.69111933595]
Centroid: CG[10.93988888889; 4.23303977865]
Coordinates of the circumscribed circle: U[15; 3.32990801584]
Coordinates of the inscribed circle: I[6.5; 5.21655589149]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.9733349274° = 154°58'24″ = 0.43767974559 rad
∠ B' = β' = 102.5133325363° = 102°30'48″ = 1.35223975988 rad
∠ C' = γ' = 102.5133325363° = 102°30'48″ = 1.35223975988 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 = 30 ; ; c = 30 ; ;

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

p = a+b+c = 13+30+30 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-13)(36.5-30)(36.5-30) } ; ; T = sqrt{ 36239.94 } = 190.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 190.37 }{ 13 } = 29.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 190.37 }{ 30 } = 12.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 190.37 }{ 30 } = 12.69 ; ;

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**2-30**2-30**2 }{ 2 * 30 * 30 } ) = 25° 1'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 30**2-13**2-30**2 }{ 2 * 13 * 30 } ) = 77° 29'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-13**2-30**2 }{ 2 * 30 * 13 } ) = 77° 29'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 190.37 }{ 36.5 } = 5.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 25° 1'36" } = 15.36 ; ;




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