4 13 13 triangle

Acute isosceles triangle.

Sides: a = 4   b = 13   c = 13

Area: T = 25.69904651573
Perimeter: p = 30
Semiperimeter: s = 15

Angle ∠ A = α = 17.76997661969° = 17°41'59″ = 0.3098919197 rad
Angle ∠ B = β = 81.15501169016° = 81°9' = 1.41663367283 rad
Angle ∠ C = γ = 81.15501169016° = 81°9' = 1.41663367283 rad

Height: ha = 12.84552325787
Height: hb = 3.9522379255
Height: hc = 3.9522379255

Median: ma = 12.84552325787
Median: mb = 7.08987234394
Median: mc = 7.08987234394

Inradius: r = 1.71326976772
Circumradius: R = 6.57883160782

Vertex coordinates: A[13; 0] B[0; 0] C[0.61553846154; 3.9522379255]
Centroid: CG[4.53884615385; 1.31774597517]
Coordinates of the circumscribed circle: U[6.5; 1.01220486274]
Coordinates of the inscribed circle: I[2; 1.71326976772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.3300233803° = 162°18'1″ = 0.3098919197 rad
∠ B' = β' = 98.85498830984° = 98°51' = 1.41663367283 rad
∠ C' = γ' = 98.85498830984° = 98°51' = 1.41663367283 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 = 4 ; ; b = 13 ; ; c = 13 ; ;

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

p = a+b+c = 4+13+13 = 30 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30 }{ 2 } = 15 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15 * (15-4)(15-13)(15-13) } ; ; T = sqrt{ 660 } = 25.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.69 }{ 4 } = 12.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.69 }{ 13 } = 3.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.69 }{ 13 } = 3.95 ; ;

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{ 4**2-13**2-13**2 }{ 2 * 13 * 13 } ) = 17° 41'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-4**2-13**2 }{ 2 * 4 * 13 } ) = 81° 9' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-4**2-13**2 }{ 2 * 13 * 4 } ) = 81° 9' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.69 }{ 15 } = 1.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 17° 41'59" } = 6.58 ; ;




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