Calculate dimensions of an inscribed square
Step 1: Identify the Circle Radius
Find r, the distance from the circle’s center to the circumference.
Why: The radius determines the diagonal of the inscribed square.
Step 2: Understand Geometric Constraint
Recognize that the square is inscribed with all four vertices on the circle.
Why: The diagonal of the square equals the diameter of the circle: diagonal = 2r.
Step 3: Relate Diagonal to Side Using 45-45-90 Triangle
The diagonal d = 2r forms a 45-45-90 triangle with the sides.
Why: In a square, the diagonal creates two 45-45-90 right triangles.
Step 4: Calculate Side Length
Use s = diagonal / √2 = 2r / √2 = r√2.
Why: In a 45-45-90 triangle, the hypotenuse is √2 times each leg (side).
Step 5: Calculate and Verify Area
Square area A = s² = (r√2)² = 2r². Verify it’s less than the circle area (πr²).
Why: The inscribed square must have less area than its enclosing circle.
When a square is inscribed in a circle, its diagonal is equal to the diameter of the circle.s = r√2
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