Isosceles triangle calculator (a)

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 angle γ.

Acute isosceles triangle.

Sides: a = 32   b = 32   c = 24.49217396714

Area: T = 362.0398671968
Perimeter: p = 88.49217396714
Semiperimeter: s = 44.24658698357

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 22.6277416998
Height: hb = 22.6277416998
Height: hc = 29.56441450404

Median: ma = 23.57880121313
Median: mb = 23.57880121313
Median: mc = 29.56441450404

Inradius: r = 8.18224286269
Circumradius: R = 17.31882752047

Vertex coordinates: A[24.49217396714; 0] B[0; 0] C[12.24658698357; 29.56441450404]
Centroid: CG[12.24658698357; 9.85547150135]
Coordinates of the circumscribed circle: U[12.24658698357; 12.24658698357]
Coordinates of the inscribed circle: I[12.24658698357; 8.18224286269]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 135° = 0.78553981634 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 = 32 ; ; b = 32 ; ; c = 24.49 ; ;

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

p = a+b+c = 32+32+24.49 = 88.49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 88.49 }{ 2 } = 44.25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 44.25 * (44.25-32)(44.25-32)(44.25-24.49) } ; ; T = sqrt{ 131072 } = 362.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 362.04 }{ 32 } = 22.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 362.04 }{ 32 } = 22.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 362.04 }{ 24.49 } = 29.56 ; ;

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{ 32**2+24.49**2-32**2 }{ 2 * 32 * 24.49 } ) = 67° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 32**2+24.49**2-32**2 }{ 2 * 32 * 24.49 } ) = 67° 30' ; ; gamma = 180° - alpha - beta = 180° - 67° 30' - 67° 30' = 45° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 362.04 }{ 44.25 } = 8.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 32 }{ 2 * sin 67° 30' } = 17.32 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 24.49**2 - 32**2 } }{ 2 } = 23.578 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24.49**2+2 * 32**2 - 32**2 } }{ 2 } = 23.578 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 32**2 - 24.49**2 } }{ 2 } = 29.564 ; ;
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