Isosceles triangle calculator (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 γ.

Acute isosceles triangle.

Sides: a = 3.82879286376   b = 3.82879286376   c = 4.5

Area: T = 6.96879334724
Perimeter: p = 12.15658572752
Semiperimeter: s = 6.07879286376

Angle ∠ A = α = 54° = 0.94224777961 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 72° = 1.25766370614 rad

Height: ha = 3.64105764747
Height: hb = 3.64105764747
Height: hc = 3.09768593211

Median: ma = 3.71332545581
Median: mb = 3.71332545581
Median: mc = 3.09768593211

Inradius: r = 1.14664322614
Circumradius: R = 2.36657900045

Vertex coordinates: A[4.5; 0] B[0; 0] C[2.25; 3.09768593211]
Centroid: CG[2.25; 1.03222864404]
Coordinates of the circumscribed circle: U[2.25; 0.73110693165]
Coordinates of the inscribed circle: I[2.25; 1.14664322614]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126° = 0.94224777961 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 108° = 1.25766370614 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 = 3.83 ; ; b = 3.83 ; ; c = 4.5 ; ;

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

p = a+b+c = 3.83+3.83+4.5 = 12.16 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.16 }{ 2 } = 6.08 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.08 * (6.08-3.83)(6.08-3.83)(6.08-4.5) } ; ; T = sqrt{ 48.55 } = 6.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.97 }{ 3.83 } = 3.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.97 }{ 3.83 } = 3.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.97 }{ 4.5 } = 3.1 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 3.83**2-3.83**2-4.5**2 }{ 2 * 3.83 * 4.5 } ) = 54° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.83**2-3.83**2-4.5**2 }{ 2 * 3.83 * 4.5 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.5**2-3.83**2-3.83**2 }{ 2 * 3.83 * 3.83 } ) = 72° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.97 }{ 6.08 } = 1.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.83 }{ 2 * sin 54° } = 2.37 ; ;




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