Isosceles triangle calculator (a,b)

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 b.

Acute isosceles triangle.

Sides: a = 45   b = 45   c = 30

Area: T = 636.3966103068
Perimeter: p = 120
Semiperimeter: s = 60

Angle ∠ A = α = 70.52987793655° = 70°31'44″ = 1.23109594173 rad
Angle ∠ B = β = 70.52987793655° = 70°31'44″ = 1.23109594173 rad
Angle ∠ C = γ = 38.9422441269° = 38°56'33″ = 0.68796738189 rad

Height: ha = 28.28442712475
Height: hb = 28.28442712475
Height: hc = 42.42664068712

Median: ma = 30.92332921921
Median: mb = 30.92332921921
Median: mc = 42.42664068712

Inradius: r = 10.60766017178
Circumradius: R = 23.8654853865

Vertex coordinates: A[30; 0] B[0; 0] C[15; 42.42664068712]
Centroid: CG[15; 14.14221356237]
Coordinates of the circumscribed circle: U[15; 18.56215530061]
Coordinates of the inscribed circle: I[15; 10.60766017178]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 109.4711220634° = 109°28'16″ = 1.23109594173 rad
∠ B' = β' = 109.4711220634° = 109°28'16″ = 1.23109594173 rad
∠ C' = γ' = 141.0587558731° = 141°3'27″ = 0.68796738189 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 = 45 ; ; b = 45 ; ; c = 30 ; ;

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

p = a+b+c = 45+45+30 = 120 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 120 }{ 2 } = 60 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60 * (60-45)(60-45)(60-30) } ; ; T = sqrt{ 405000 } = 636.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 636.4 }{ 45 } = 28.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 636.4 }{ 45 } = 28.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 636.4 }{ 30 } = 42.43 ; ;

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{ 45**2+30**2-45**2 }{ 2 * 45 * 30 } ) = 70° 31'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+30**2-45**2 }{ 2 * 45 * 30 } ) = 70° 31'44" ; ; gamma = 180° - alpha - beta = 180° - 70° 31'44" - 70° 31'44" = 38° 56'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 636.4 }{ 60 } = 10.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 70° 31'44" } = 23.86 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 30**2 - 45**2 } }{ 2 } = 30.923 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 45**2 - 45**2 } }{ 2 } = 30.923 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 45**2 - 30**2 } }{ 2 } = 42.426 ; ;
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