Isosceles triangle calculator (p)

Please enter two properties of the isosceles triangle

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


You have entered perimeter p and angle γ.

Right isosceles triangle.

Sides: a = 79.37440622984   b = 79.37440622984   c = 112.2521875403

Area: T = 3150.121088288
Perimeter: p = 271
Semiperimeter: s = 135.5

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

Height: ha = 79.37440622984
Height: hb = 79.37440622984
Height: hc = 56.12659377016

Median: ma = 88.74328994748
Median: mb = 88.74328994748
Median: mc = 56.12659377016

Inradius: r = 23.24881245969
Circumradius: R = 56.12659377016

Vertex coordinates: A[112.2521875403; 0] B[0; 0] C[56.12659377016; 56.12659377016]
Centroid: CG[56.12659377016; 18.70986459005]
Coordinates of the circumscribed circle: U[56.12659377016; 0]
Coordinates of the inscribed circle: I[56.12659377016; 23.24881245969]

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 = 79.37 ; ; b = 79.37 ; ; c = 112.25 ; ;

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

p = a+b+c = 79.37+79.37+112.25 = 271 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 271 }{ 2 } = 135.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.5 * (135.5-79.37)(135.5-79.37)(135.5-112.25) } ; ; T = sqrt{ 9923261.58 } = 3150.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3150.12 }{ 79.37 } = 79.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3150.12 }{ 79.37 } = 79.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3150.12 }{ 112.25 } = 56.13 ; ;

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{ 79.37**2+112.25**2-79.37**2 }{ 2 * 79.37 * 112.25 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 79.37**2+112.25**2-79.37**2 }{ 2 * 79.37 * 112.25 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3150.12 }{ 135.5 } = 23.25 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 79.37 }{ 2 * sin 45° } = 56.13 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 79.37**2+2 * 112.25**2 - 79.37**2 } }{ 2 } = 88.743 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 112.25**2+2 * 79.37**2 - 79.37**2 } }{ 2 } = 88.743 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 79.37**2+2 * 79.37**2 - 112.25**2 } }{ 2 } = 56.126 ; ;
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