457.3 457.3 457.3 triangle

Equilateral triangle.

Sides: a = 457.3   b = 457.3   c = 457.3

Area: T = 90553.04108315
Perimeter: p = 1371.9
Semiperimeter: s = 685.95

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

Height: ha = 396.0333417151
Height: hb = 396.0333417151
Height: hc = 396.0333417151

Median: ma = 396.0333417151
Median: mb = 396.0333417151
Median: mc = 396.0333417151

Inradius: r = 132.011113905
Circumradius: R = 264.02222781

Vertex coordinates: A[457.3; 0] B[0; 0] C[228.65; 396.0333417151]
Centroid: CG[228.65; 132.011113905]
Coordinates of the circumscribed circle: U[228.65; 132.011113905]
Coordinates of the inscribed circle: I[228.65; 132.011113905]

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?

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 = 457.3 ; ; b = 457.3 ; ; c = 457.3 ; ;

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

p = a+b+c = 457.3+457.3+457.3 = 1371.9 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1371.9 }{ 2 } = 685.95 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 685.95 * (685.95-457.3)(685.95-457.3)(685.95-457.3) } ; ; T = sqrt{ 8199853203.83 } = 90553.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90553.04 }{ 457.3 } = 396.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90553.04 }{ 457.3 } = 396.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90553.04 }{ 457.3 } = 396.03 ; ;

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{ 457.3**2+457.3**2-457.3**2 }{ 2 * 457.3 * 457.3 } ) = 60° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 457.3**2+457.3**2-457.3**2 }{ 2 * 457.3 * 457.3 } ) = 60° ; ;
 gamma = 180° - alpha - beta = 180° - 60° - 60° = 60° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90553.04 }{ 685.95 } = 132.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 457.3 }{ 2 * sin 60° } = 264.02 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 457.3**2+2 * 457.3**2 - 457.3**2 } }{ 2 } = 396.033 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 457.3**2+2 * 457.3**2 - 457.3**2 } }{ 2 } = 396.033 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 457.3**2+2 * 457.3**2 - 457.3**2 } }{ 2 } = 396.033 ; ;
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