Isosceles triangle calculator (h)

Please enter two properties of the isosceles triangle

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


You have entered height hc and angle γ.

Acute isosceles triangle.

Sides: a = 31.94546410072   b = 31.94546410072   c = 24.44993697326

Area: T = 360.787712446
Perimeter: p = 88.33986517471
Semiperimeter: s = 44.16993258735

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 22.58882722788
Height: hb = 22.58882722788
Height: hc = 29.513

Median: ma = 23.53772229125
Median: mb = 23.53772229125
Median: mc = 29.513

Inradius: r = 8.1688273283
Circumradius: R = 17.28883151337

Vertex coordinates: A[24.44993697326; 0] B[0; 0] C[12.22546848663; 29.513]
Centroid: CG[12.22546848663; 9.83876666667]
Coordinates of the circumscribed circle: U[12.22546848663; 12.22546848663]
Coordinates of the inscribed circle: I[12.22546848663; 8.1688273283]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

1. Input data entered: angle γ and height hc

 gamma = 45° ; ; hc = 29.513 ; ;

2. From we calculate side b:

b = a = 31.945 ; ;

3. From height h we calculate side c - Pythagorean theorem:

a**2 = (c/2)**2 + h**2 ; ; ; ; c = 2 sqrt{ a**2-h**2 } = 2 * sqrt{ 31.945**2-29.513**2 } = 24.449 ; ;

4. From side c we calculate perimeter p:

p = 2a + c = 2 * 31.945 + 24.449 = 88.339 ; ;
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 = 31.94 ; ; b = 31.94 ; ; c = 24.45 ; ;

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

p = a+b+c = 31.94+31.94+24.45 = 88.34 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 88.34 }{ 2 } = 44.17 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 44.17 * (44.17-31.94)(44.17-31.94)(44.17-24.45) } ; ; T = sqrt{ 130167.35 } = 360.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 360.79 }{ 31.94 } = 22.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 360.79 }{ 31.94 } = 22.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 360.79 }{ 24.45 } = 29.51 ; ;

9. 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{ 31.94**2+24.45**2-31.94**2 }{ 2 * 31.94 * 24.45 } ) = 67° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31.94**2+24.45**2-31.94**2 }{ 2 * 31.94 * 24.45 } ) = 67° 30' ; ; gamma = 180° - alpha - beta = 180° - 67° 30' - 67° 30' = 45° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 360.79 }{ 44.17 } = 8.17 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 31.94 }{ 2 * sin 67° 30' } = 17.29 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.94**2+2 * 24.45**2 - 31.94**2 } }{ 2 } = 23.537 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24.45**2+2 * 31.94**2 - 31.94**2 } }{ 2 } = 23.537 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.94**2+2 * 31.94**2 - 24.45**2 } }{ 2 } = 29.513 ; ;
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