Isosceles triangle calculator (A,B,p)

Please enter two properties of the isosceles triangle

Use symbols: a, b, h, T, p, A, B, C, r, R


You have entered perimeter p, angle α and angle β.

Acute isosceles triangle.

Sides: a = 8.39902216693   b = 8.39902216693   c = 4.06595566615

Area: T = 16.52444177346
Perimeter: p = 20.84
Semiperimeter: s = 10.42

Angle ∠ A = α = 76° = 1.32664502315 rad
Angle ∠ B = β = 76° = 1.32664502315 rad
Angle ∠ C = γ = 28° = 0.48986921906 rad

Height: ha = 3.93989704792
Height: hb = 3.93989704792
Height: hc = 8.14109962282

Median: ma = 5.08332032282
Median: mb = 5.08332032282
Median: mc = 8.14109962282

Inradius: r = 1.58658366348
Circumradius: R = 4.32435384028

Vertex coordinates: A[4.06595566615; 0] B[0; 0] C[2.03297783307; 8.14109962282]
Centroid: CG[2.03297783307; 2.71436654094]
Coordinates of the circumscribed circle: U[2.03297783307; 3.81774578254]
Coordinates of the inscribed circle: I[2.03297783307; 1.58658366348]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 104° = 1.32664502315 rad
∠ B' = β' = 104° = 1.32664502315 rad
∠ C' = γ' = 152° = 0.48986921906 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 = 8.39 ; ; b = 8.39 ; ; c = 4.06 ; ;

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

p = a+b+c = 8.39+8.39+4.06 = 20.84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.84 }{ 2 } = 10.42 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.42 * (10.42-8.39)(10.42-8.39)(10.42-4.06) } ; ; T = sqrt{ 273.06 } = 16.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.52 }{ 8.39 } = 3.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.52 }{ 8.39 } = 3.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.52 }{ 4.06 } = 8.14 ; ;

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{ 8.39**2-8.39**2-4.06**2 }{ 2 * 8.39 * 4.06 } ) = 76° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.39**2-8.39**2-4.06**2 }{ 2 * 8.39 * 4.06 } ) = 76° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.06**2-8.39**2-8.39**2 }{ 2 * 8.39 * 8.39 } ) = 28° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.52 }{ 10.42 } = 1.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.39 }{ 2 * sin 76° } = 4.32 ; ;




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