Isosceles triangle calculator (a,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 a and b.

Acute isosceles triangle.

Sides: a = 500   b = 500   c = 4.5

Area: T = 1124.989860932
Perimeter: p = 1004.5
Semiperimeter: s = 502.25

Angle ∠ A = α = 89.7422168122° = 89°44'32″ = 1.56662963116 rad
Angle ∠ B = β = 89.7422168122° = 89°44'32″ = 1.56662963116 rad
Angle ∠ C = γ = 0.5165663756° = 0°30'56″ = 0.00990000304 rad

Height: ha = 4.54999544373
Height: hb = 4.54999544373
Height: hc = 499.9954937474

Median: ma = 250.022024918
Median: mb = 250.022024918
Median: mc = 499.9954937474

Inradius: r = 2.24398976791
Circumradius: R = 250.0032531288

Vertex coordinates: A[4.5; 0] B[0; 0] C[2.25; 499.9954937474]
Centroid: CG[2.25; 166.6654979158]
Coordinates of the circumscribed circle: U[2.25; 249.9922406186]
Coordinates of the inscribed circle: I[2.25; 2.24398976791]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90.2587831878° = 90°15'28″ = 1.56662963116 rad
∠ B' = β' = 90.2587831878° = 90°15'28″ = 1.56662963116 rad
∠ C' = γ' = 179.4844336244° = 179°29'4″ = 0.00990000304 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 = 500 ; ; b = 500 ; ; c = 4.5 ; ;

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

p = a+b+c = 500+500+4.5 = 1004.5 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1004.5 }{ 2 } = 502.25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 502.25 * (502.25-500)(502.25-500)(502.25-4.5) } ; ; T = sqrt{ 1265599.37 } = 1124.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1124.99 }{ 500 } = 4.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1124.99 }{ 500 } = 4.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1124.99 }{ 4.5 } = 499.99 ; ;

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{ 500**2+4.5**2-500**2 }{ 2 * 500 * 4.5 } ) = 89° 44'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 500**2+4.5**2-500**2 }{ 2 * 500 * 4.5 } ) = 89° 44'32" ; ; gamma = 180° - alpha - beta = 180° - 89° 44'32" - 89° 44'32" = 0° 30'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1124.99 }{ 502.25 } = 2.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 500 }{ 2 * sin 89° 44'32" } = 250 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 500**2+2 * 4.5**2 - 500**2 } }{ 2 } = 250.02 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.5**2+2 * 500**2 - 500**2 } }{ 2 } = 250.02 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 500**2+2 * 500**2 - 4.5**2 } }{ 2 } = 499.995 ; ;
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