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.

Right isosceles triangle.

Sides: a = 63.64396103068   b = 63.64396103068   c = 90

Area: T = 2025
Perimeter: p = 217.2799220614
Semiperimeter: s = 108.6439610307

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 63.64396103068
Height: hb = 63.64396103068
Height: hc = 45

Median: ma = 71.15112473538
Median: mb = 71.15112473538
Median: mc = 45

Inradius: r = 18.64396103068
Circumradius: R = 45

Vertex coordinates: A[90; 0] B[0; 0] C[45; 45]
Centroid: CG[45; 15]
Coordinates of the circumscribed circle: U[45; -0]
Coordinates of the inscribed circle: I[45; 18.64396103068]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 63.64 ; ; b = 63.64 ; ; c = 90 ; ;

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

p = a+b+c = 63.64+63.64+90 = 217.28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 217.28 }{ 2 } = 108.64 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 108.64 * (108.64-63.64)(108.64-63.64)(108.64-90) } ; ; T = sqrt{ 4100625 } = 2025 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2025 }{ 63.64 } = 63.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2025 }{ 63.64 } = 63.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2025 }{ 90 } = 45 ; ;

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{ 63.64**2+90**2-63.64**2 }{ 2 * 63.64 * 90 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 63.64**2+90**2-63.64**2 }{ 2 * 63.64 * 90 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2025 }{ 108.64 } = 18.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 63.64 }{ 2 * sin 45° } = 45 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 63.64**2+2 * 90**2 - 63.64**2 } }{ 2 } = 71.151 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 63.64**2 - 63.64**2 } }{ 2 } = 71.151 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 63.64**2+2 * 63.64**2 - 90**2 } }{ 2 } = 45 ; ;
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