Isosceles triangle calculator (b)

Please enter two properties of the isosceles triangle

Use symbols: a, b, h, T, p, A, B, C, r, R


You have entered side b and angle γ.

Acute isosceles triangle.

Sides: a = 56.40658411448   b = 56.40658411448   c = 42.26

Area: T = 1105.069910554
Perimeter: p = 155.0721682289
Semiperimeter: s = 77.53658411448

Angle ∠ A = α = 68° = 1.18768238914 rad
Angle ∠ B = β = 68° = 1.18768238914 rad
Angle ∠ C = γ = 44° = 0.76879448709 rad

Height: ha = 39.1832789694
Height: hb = 39.1832789694
Height: hc = 52.29985852127

Median: ma = 41.09896401641
Median: mb = 41.09896401641
Median: mc = 52.29985852127

Inradius: r = 14.25223649609
Circumradius: R = 30.41878296823

Vertex coordinates: A[42.26; 0] B[0; 0] C[21.13; 52.29985852127]
Centroid: CG[21.13; 17.43328617376]
Coordinates of the circumscribed circle: U[21.13; 21.88107555304]
Coordinates of the inscribed circle: I[21.13; 14.25223649609]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112° = 1.18768238914 rad
∠ B' = β' = 112° = 1.18768238914 rad
∠ C' = γ' = 136° = 0.76879448709 rad

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How did we calculate this triangle?

1. Input data entered: side b and angle γ

b = 42.26 ; ; gamma = 44° ; ;

2. From we calculate side a - Pythagorean theorem:

a**2 = h**2 + (c/2)**2 ; ; a = sqrt{ h**2 + (c/2)**2 } = sqrt{ 52.299**2 + (42.26/2)**2 } = 56.406 ; ;

3. From side a we calculate perimeter p:

p = 2a + c = 2 * 56.406 + 42.26 = 155.072 ; ;

4. From side a we calculate side b:

b = a = 56.406 ; ;
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 = 56.41 ; ; b = 56.41 ; ; c = 42.26 ; ;

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

p = a+b+c = 56.41+56.41+42.26 = 155.07 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 155.07 }{ 2 } = 77.54 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77.54 * (77.54-56.41)(77.54-56.41)(77.54-42.26) } ; ; T = sqrt{ 1221177.73 } = 1105.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1105.07 }{ 56.41 } = 39.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1105.07 }{ 56.41 } = 39.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1105.07 }{ 42.26 } = 52.3 ; ;

9. 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{ 56.41**2+42.26**2-56.41**2 }{ 2 * 56.41 * 42.26 } ) = 68° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 56.41**2+42.26**2-56.41**2 }{ 2 * 56.41 * 42.26 } ) = 68° ; ; gamma = 180° - alpha - beta = 180° - 68° - 68° = 44° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1105.07 }{ 77.54 } = 14.25 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 56.41 }{ 2 * sin 68° } = 30.42 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56.41**2+2 * 42.26**2 - 56.41**2 } }{ 2 } = 41.09 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.26**2+2 * 56.41**2 - 56.41**2 } }{ 2 } = 41.09 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56.41**2+2 * 56.41**2 - 42.26**2 } }{ 2 } = 52.299 ; ;
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