Why does the square diagonal equal the circle diameter?

For a square inscribed in a circle, all four vertices of the square lie on the circle. The diagonal of the square connects two opposite vertices, which are the farthest points apart, so the diagonal must equal the circle diameter.

How do I derive the formula s = √2 × r?

Since the diagonal d = 2r and the diagonal of a square is d = s√2, we have s√2 = 2r, so s = 2r/√2 = √2 × r.

What is the relationship between square area and circle area?

Square area A_square = 2r², while circle area A_circle = πr². The ratio is A_square/A_circle = 2r²/(πr²) = 2/π ≈ 0.6366, meaning the square covers about 63.66% of the circle area.

Square in a Circle Calculator

Calculate the side length and area of a square inscribed in a circle. This calculator provides formulas for the relationship between the square diagonal and the circle diameter.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

Square in a Circle Calculator

Radius (r)

How It Works

For a square inscribed in a circle, the diagonal of the square equals the diameter of the circle: d_square = 2r.

Side length formula: s = √2 × r = d/√2, where d is the diagonal (which equals 2r).

Area formula: A = s² = 2r², since s = √2 × r, so s² = (√2 × r)² = 2r².

The relationship shows that the square area is always twice the square of the radius.

The square is the largest rectangle that can be inscribed in a circle.

Step-by-step Examples

Example 1: Basic Calculation

Calculate the side length and area of a square inscribed in a circle with radius 5 units.

Given: r = 5
Diagonal of square equals circle diameter: d = 2r = 2 × 5 = 10
Side length: s = d/√2 = 10/√2 = 5√2 ≈ 7.07 units
Area: A = s² = (5√2)² = 50 square units

Example 2: Alternative Formula

Using the direct formula s = √2 × r for a circle with radius 8.

Given: r = 8
Side length: s = √2 × r = √2 × 8 = 8√2 ≈ 11.31 units
Area: A = 2r² = 2 × 8² = 2 × 64 = 128 square units

Common Mistakes

Confusing the square diagonal with the side length - remember the diagonal equals the circle diameter.
Using the wrong formula - remember s = √2 × r, not s = r or s = 2r.
Forgetting to square the side length when calculating area: A = s², not A = s.
Not recognizing that the diagonal of the inscribed square equals the circle diameter.

Use Cases

Engineering: Designing square components that fit within circular boundaries.
Architecture: Planning square features within circular spaces.
Mathematics: Solving geometry problems involving inscribed shapes.
Design: Creating square elements that fit perfectly in circular layouts.
Manufacturing: Calculating dimensions for square parts in circular assemblies.

Frequently Asked Questions

Common questions about squares inscribed in circles.

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