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 = 5.35991000926   b = 5.35991000926   c = 6.3

Area: T = 13.65771496059
Perimeter: p = 17.01882001852
Semiperimeter: s = 8.50991000926

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

Height: ha = 5.09768070646
Height: hb = 5.09768070646
Height: hc = 4.33656030495

Median: ma = 5.19985563814
Median: mb = 5.19985563814
Median: mc = 4.33656030495

Inradius: r = 1.60550051659
Circumradius: R = 3.31221060064

Vertex coordinates: A[6.3; 0] B[0; 0] C[3.15; 4.33656030495]
Centroid: CG[3.15; 1.44552010165]
Coordinates of the circumscribed circle: U[3.15; 1.02334970431]
Coordinates of the inscribed circle: I[3.15; 1.60550051659]

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 = 5.36 ; ; b = 5.36 ; ; c = 6.3 ; ;

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

p = a+b+c = 5.36+5.36+6.3 = 17.02 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.02 }{ 2 } = 8.51 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.51 * (8.51-5.36)(8.51-5.36)(8.51-6.3) } ; ; T = sqrt{ 186.52 } = 13.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.66 }{ 5.36 } = 5.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.66 }{ 5.36 } = 5.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.66 }{ 6.3 } = 4.34 ; ;

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{ 5.36**2+6.3**2-5.36**2 }{ 2 * 5.36 * 6.3 } ) = 54° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.36**2+6.3**2-5.36**2 }{ 2 * 5.36 * 6.3 } ) = 54° ; ; gamma = 180° - alpha - beta = 180° - 54° - 54° = 72° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.66 }{ 8.51 } = 1.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.36 }{ 2 * sin 54° } = 3.31 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.36**2+2 * 6.3**2 - 5.36**2 } }{ 2 } = 5.199 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.3**2+2 * 5.36**2 - 5.36**2 } }{ 2 } = 5.199 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.36**2+2 * 5.36**2 - 6.3**2 } }{ 2 } = 4.336 ; ;
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