Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered height ha, height hb and height hc.

Equilateral triangle.

Sides: a = 3.46441016151   b = 3.46441016151   c = 3.46441016151

Area: T = 5.19661524227
Perimeter: p = 10.39223048454
Semiperimeter: s = 5.19661524227

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 3
Height: hb = 3
Height: hc = 3

Median: ma = 3
Median: mb = 3
Median: mc = 3

Inradius: r = 1
Circumradius: R = 2

Vertex coordinates: A[3.46441016151; 0] B[0; 0] C[1.73220508076; 3]
Centroid: CG[1.73220508076; 1]
Coordinates of the circumscribed circle: U[1.73220508076; 1]
Coordinates of the inscribed circle: I[1.73220508076; 1]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

1. Input data entered: height ha, height hb and height hc.

h_a = 3 ; ; h_b = 3 ; ; h_c = 3 ; ;

2. From height ha, height hb and height hc we calculate area T:

h = fraction{ fraction{ 1 }{ h_a } + fraction{ 1 }{ h_b } + fraction{ 1 }{ h_c } }{ 2 } = (1 / 3 + 1 / 3 + 1 / 3) / 2 = 0.5 ; ; ; ; S_1 = 4 * sqrt{ h * (h - 1 / h_a) * (h - 1 / h_b) * (h - 1 / h_c) } ; ; ; ; S_1 = 4 * sqrt{ 0.5 * (0.5 - 1 / 3) * (0.5 - 1 / 3) * (0.5 - 1 / 3) } = 0.192 ; ; ; ; T = fraction{ 1 }{ S_1 } = fraction{ 1 }{ 0.192 } = 5.196 ; ;

3. From area T and height ha we calculate side a:

T = fraction{ a h_a }{ 2 } ; ; ; ; a = fraction{ 2 T }{ h_a } = fraction{ 2 * 5.196 }{ 3 } = 3.464 ; ;

4. From area T and height hb we calculate side b:

T = fraction{ b h_b }{ 2 } ; ; ; ; b = fraction{ 2 T }{ h_b } = fraction{ 2 * 5.196 }{ 3 } = 3.464 ; ;

5. From area T and height hc we calculate side c:

T = fraction{ c h_c }{ 2 } ; ; ; ; c = fraction{ 2 T }{ h_c } = fraction{ 2 * 5.196 }{ 3 } = 3.464 ; ;
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 = 3.46 ; ; b = 3.46 ; ; c = 3.46 ; ;

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

p = a+b+c = 3.46+3.46+3.46 = 10.39 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.39 }{ 2 } = 5.2 ; ;

8. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.2 * (5.2-3.46)(5.2-3.46)(5.2-3.46) } ; ; T = sqrt{ 27 } = 5.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.2 }{ 3.46 } = 3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.2 }{ 3.46 } = 3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.2 }{ 3.46 } = 3 ; ;

10. 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{ 3.46**2+3.46**2-3.46**2 }{ 2 * 3.46 * 3.46 } ) = 60° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.46**2+3.46**2-3.46**2 }{ 2 * 3.46 * 3.46 } ) = 60° ; ; gamma = 180° - alpha - beta = 180° - 60° - 60° = 60° ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.2 }{ 5.2 } = 1 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.46 }{ 2 * sin 60° } = 2 ; ;

13. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.46**2+2 * 3.46**2 - 3.46**2 } }{ 2 } = 3 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.46**2+2 * 3.46**2 - 3.46**2 } }{ 2 } = 3 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.46**2+2 * 3.46**2 - 3.46**2 } }{ 2 } = 3 ; ;
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