Isosceles triangle calculator (A,B,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, angle α, angle β and angle γ.

Acute isosceles triangle.

Sides: a = 20.5   b = 20.5   c = 12.67696967694

Area: T = 123.5088376138
Perimeter: p = 53.67696967694
Semiperimeter: s = 26.83548483847

Angle ∠ A = α = 72° = 1.25766370614 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 36° = 0.62883185307 rad

Height: ha = 12.0549597672
Height: hb = 12.0549597672
Height: hc = 19.49766585841

Median: ma = 13.61333430176
Median: mb = 13.61333430176
Median: mc = 19.49766585841

Inradius: r = 4.60325367599
Circumradius: R = 10.77774877984

Vertex coordinates: A[12.67696967694; 0] B[0; 0] C[6.33548483847; 19.49766585841]
Centroid: CG[6.33548483847; 6.49988861947]
Coordinates of the circumscribed circle: U[6.33548483847; 8.71991707856]
Coordinates of the inscribed circle: I[6.33548483847; 4.60325367599]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108° = 1.25766370614 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 144° = 0.62883185307 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 = 20.5 ; ; b = 20.5 ; ; c = 12.67 ; ;

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

p = a+b+c = 20.5+20.5+12.67 = 53.67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53.67 }{ 2 } = 26.83 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.83 * (26.83-20.5)(26.83-20.5)(26.83-12.67) } ; ; T = sqrt{ 15254.32 } = 123.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 123.51 }{ 20.5 } = 12.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 123.51 }{ 20.5 } = 12.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 123.51 }{ 12.67 } = 19.5 ; ;

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{ 20.5**2-20.5**2-12.67**2 }{ 2 * 20.5 * 12.67 } ) = 72° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20.5**2-20.5**2-12.67**2 }{ 2 * 20.5 * 12.67 } ) = 72° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.67**2-20.5**2-20.5**2 }{ 2 * 20.5 * 20.5 } ) = 36° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 123.51 }{ 26.83 } = 4.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.5 }{ 2 * sin 72° } = 10.78 ; ;




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