Isosceles triangle calculator (A,c)

Please enter two properties of the isosceles triangle

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


You have entered side c and angle α.

Acute isosceles triangle.

Sides: a = 70.00875722087   b = 70.00875722087   c = 90

Area: T = 2413.3011025
Perimeter: p = 230.0155144417
Semiperimeter: s = 115.0087572209

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 68.94439998807
Height: hb = 68.94439998807
Height: hc = 53.62989116667

Median: ma = 72.63110198306
Median: mb = 72.63110198306
Median: mc = 53.62989116667

Inradius: r = 20.9843844617
Circumradius: R = 45.69441975349

Vertex coordinates: A[90; 0] B[0; 0] C[45; 53.62989116667]
Centroid: CG[45; 17.87663038889]
Coordinates of the circumscribed circle: U[45; 7.93547141319]
Coordinates of the inscribed circle: I[45; 20.9843844617]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 100° = 1.39662634016 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 = 70.01 ; ; b = 70.01 ; ; c = 90 ; ;

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

p = a+b+c = 70.01+70.01+90 = 230.02 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 230.02 }{ 2 } = 115.01 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 115.01 * (115.01-70.01)(115.01-70.01)(115.01-90) } ; ; T = sqrt{ 5824021.84 } = 2413.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2413.3 }{ 70.01 } = 68.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2413.3 }{ 70.01 } = 68.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2413.3 }{ 90 } = 53.63 ; ;

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{ 70.01**2+90**2-70.01**2 }{ 2 * 70.01 * 90 } ) = 50° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 70.01**2+90**2-70.01**2 }{ 2 * 70.01 * 90 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 50° - 50° = 80° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2413.3 }{ 115.01 } = 20.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 70.01 }{ 2 * sin 50° } = 45.69 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.01**2+2 * 90**2 - 70.01**2 } }{ 2 } = 72.631 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 70.01**2 - 70.01**2 } }{ 2 } = 72.631 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.01**2+2 * 70.01**2 - 90**2 } }{ 2 } = 53.629 ; ;
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