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.

Obtuse isosceles triangle.

Sides: a = 7.5   b = 7.5   c = 11.14771723822

Area: T = 27.97109283072
Perimeter: p = 26.14771723822
Semiperimeter: s = 13.07435861911

Angle ∠ A = α = 42° = 0.73330382858 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 96° = 1.67655160819 rad

Height: ha = 7.45989142153
Height: hb = 7.45989142153
Height: hc = 5.01884795477

Median: ma = 8.72988158452
Median: mb = 8.72988158452
Median: mc = 5.01884795477

Inradius: r = 2.13994992849
Circumradius: R = 5.6044287062

Vertex coordinates: A[11.14771723822; 0] B[0; 0] C[5.57435861911; 5.01884795477]
Centroid: CG[5.57435861911; 1.67328265159]
Coordinates of the circumscribed circle: U[5.57435861911; -0.58658075143]
Coordinates of the inscribed circle: I[5.57435861911; 2.13994992849]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138° = 0.73330382858 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 84° = 1.67655160819 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 = 7.5 ; ; b = 7.5 ; ; c = 11.15 ; ;

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

p = a+b+c = 7.5+7.5+11.15 = 26.15 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.15 }{ 2 } = 13.07 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.07 * (13.07-7.5)(13.07-7.5)(13.07-11.15) } ; ; T = sqrt{ 782.37 } = 27.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27.97 }{ 7.5 } = 7.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27.97 }{ 7.5 } = 7.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27.97 }{ 11.15 } = 5.02 ; ;

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{ 7.5**2-7.5**2-11.15**2 }{ 2 * 7.5 * 11.15 } ) = 42° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.5**2-7.5**2-11.15**2 }{ 2 * 7.5 * 11.15 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.15**2-7.5**2-7.5**2 }{ 2 * 7.5 * 7.5 } ) = 96° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27.97 }{ 13.07 } = 2.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 42° } = 5.6 ; ;




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