13 13 22 triangle

Obtuse isosceles triangle.

Sides: a = 13   b = 13   c = 22

Area: T = 76.2110235533
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 32.2044227504° = 32°12'15″ = 0.5622069803 rad
Angle ∠ B = β = 32.2044227504° = 32°12'15″ = 0.5622069803 rad
Angle ∠ C = γ = 115.5921544992° = 115°35'30″ = 2.01774530476 rad

Height: ha = 11.72546516205
Height: hb = 11.72546516205
Height: hc = 6.92882032303

Median: ma = 16.86597153001
Median: mb = 16.86597153001
Median: mc = 6.92882032303

Inradius: r = 3.17554264805
Circumradius: R = 12.19765244366

Vertex coordinates: A[22; 0] B[0; 0] C[11; 6.92882032303]
Centroid: CG[11; 2.30994010768]
Coordinates of the circumscribed circle: U[11; -5.26883212064]
Coordinates of the inscribed circle: I[11; 3.17554264805]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.7965772496° = 147°47'45″ = 0.5622069803 rad
∠ B' = β' = 147.7965772496° = 147°47'45″ = 0.5622069803 rad
∠ C' = γ' = 64.40884550079° = 64°24'30″ = 2.01774530476 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 = 13 ; ; c = 22 ; ;

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

p = a+b+c = 13+13+22 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-13)(24-13)(24-22) } ; ; T = sqrt{ 5808 } = 76.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76.21 }{ 13 } = 11.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76.21 }{ 13 } = 11.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76.21 }{ 22 } = 6.93 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13**2+22**2-13**2 }{ 2 * 13 * 22 } ) = 32° 12'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+22**2-13**2 }{ 2 * 13 * 22 } ) = 32° 12'15" ; ; gamma = 180° - alpha - beta = 180° - 32° 12'15" - 32° 12'15" = 115° 35'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76.21 }{ 24 } = 3.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13 }{ 2 * sin 32° 12'15" } = 12.2 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 22**2 - 13**2 } }{ 2 } = 16.86 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 13**2 - 13**2 } }{ 2 } = 16.86 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 13**2 - 22**2 } }{ 2 } = 6.928 ; ;
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