How do you find the circumscribed circle radius of a square?

For a square with side s, the circumscribed circle radius is r = s sqrt(2) / 2. The diagonal of the square (s sqrt(2)) equals the diameter of the circumscribed circle. For side 10: r = 10 times 1.414 / 2 = 7.071 units.

What percentage of a circle is filled by its inscribed square?

The inscribed square fills exactly 2 / pi = 63.66% of the circle area. This is constant regardless of circle size. Circle area = pi r squared. Square area = 2r squared. Ratio = 2r squared / (pi r squared) = 2 / pi = 63.66%.

What is the diagonal of a square inscribed in a circle?

The diagonal of the inscribed square is always equal to the diameter of the circle: diagonal = 2r. Each diagonal of the square passes through the center and connects two points on the circle, making it a diameter by definition.

What is the side of a square that fits exactly in a circle of radius 5?

For r = 5: side = 5 times sqrt(2) = 7.071 units. Area = 50 sq units. Perimeter = 28.284 units. Diagonal = 10 units (equals circle diameter).

What is the relationship between the areas of a square and its circumscribed circle?

Square area / Circle area = s squared / (pi times (s sqrt(2) / 2) squared) = 2 / pi = 0.6366. The square always occupies 63.66% of its circumscribed circle.

What is the circumscribed circle of a unit square?

A unit square (side = 1) has circumscribed circle radius r = sqrt(2) / 2 = 0.7071 units, diameter = sqrt(2) = 1.4142 units, area = pi times 0.5 = 1.5708 square units, and circumference = pi times sqrt(2) = 4.443 units.

Can any square be inscribed in a circle?

Yes. Every square has a unique circumscribed circle because all four vertices are equidistant from the center. The center of the circumscribed circle is the intersection of the diagonals. Conversely, every circle has exactly one largest inscribed square whose diagonal equals the diameter.

Square in a Circle Calculator

Q: What is the formula for a square inscribed in a circle?

For a square inscribed in a circle of radius r: side = r times sqrt(2). This follows from the diagonal of the square equaling the diameter (2r). Since the diagonal of a square with side s is s times sqrt(2), setting s sqrt(2) = 2r gives s = r sqrt(2). Area = 2r squared, perimeter = 4r sqrt(2).

Enter a circle radius to find the largest inscribed square, or enter a square side to find the circumscribed circle. Formulas, worked examples, and area ratios explained.

🔲 What is a Square in a Circle?

A square inscribed in a circle is the largest square that fits exactly inside a given circle, with all four corners touching the circle's boundary. The relationship between a square and its circumscribed circle is one of the most fundamental constructions in Euclidean geometry. The diagonal of the inscribed square is always equal to the diameter of the circle, because the diagonal spans the full width of the circle from one vertex to the opposite vertex, passing through the center.

This relationship appears in a wide range of practical situations. In manufacturing, a "square peg in a round hole" problem requires knowing the minimum circular hole diameter that can accept a square cross-section shaft. In woodworking, a craftsperson cutting the largest square tile from a circular disk needs the inscribed square formula. In structural engineering, a square cross-section beam inscribed in a circular tube maximizes the wood volume that can be milled from a cylindrical log. Architects designing square courtyards within circular buildings use this relationship to maximize the usable floor area.

There are two complementary ways to approach the square-circle relationship. The first approach starts with a known circle radius and finds the inscribed square. The second starts with a known square side and finds the circumscribed circle. Both use the same core formula: side = r times sqrt(2), rearranged as r = side times sqrt(2) / 2 for the reverse. The area ratio between square and circle is always 2 divided by pi = 63.66%, an elegant constant that holds for any size.

A key theorem underlying this calculator is that every square is a cyclic polygon, meaning all four of its vertices lie on a common circle. The center of this circumscribed circle is the intersection of the diagonals of the square. The diagonals of a square bisect each other at right angles, and each diagonal is a diameter of the circumscribed circle. By Thales' theorem, any angle inscribed in a semicircle and subtending the diameter is a right angle, confirming that the four corners of the square each make a 90-degree angle when viewed from opposite vertices across the diagonal.

📐 Formulas

Inscribed Square s = r × √2

s = side length of the inscribed square

r = radius of the circle

Square area = s² = 2r²

Square perimeter = 4s = 4r√2

Diagonal = s√2 = 2r (equals circle diameter)

Example: Circle radius r = 10; s = 10 × 1.4142 = 14.142 units; area = 200 sq units

Circumscribed Circle r = s√2 ÷ 2

r = radius of the circumscribed circle

s = side length of the square

Circle area = πr² = πs² ÷ 2

Circumference = 2πr = πs√2

Example: Square side s = 8; r = 8 × 1.4142 / 2 = 5.657 units; circle area = 100.53 sq units

Area Ratio Square ÷ Circle = 2 ÷ π ≈ 63.66%

Square area = 2r²

Circle area = πr²

Ratio = 2r² ÷ (πr²) = 2 ÷ π = 0.6366 (constant for any radius)

Interpretation: The inscribed square always covers 63.66% of the enclosing circle, no matter how large or small.

📖 How to Use This Calculator

Steps

Choose your input type - Select Inscribed Square if you know the circle radius and want the square dimensions. Select Circumscribed Circle if you know the square side and want the enclosing circle.

Enter the measurement - Type the radius (Inscribed Square mode) or the side length (Circumscribed Circle mode) into the input field. The slider updates automatically and provides quick coarse adjustment.

Click Calculate - Press Calculate to see all results: side, area, perimeter, diagonal, circle area, and the square-to-circle area ratio showing 63.66%.

Use the results - Copy the result to the clipboard, print the page, or use the Copy link button to save a permalink with your exact inputs for sharing or future reference.

💡 Example Calculations

Example 1 - Largest Square Tile from a Circular Disk (r = 15 cm)

A woodworker has a circular disk with radius 15 cm and wants the largest square piece that can be cut from it.

Circle radius r = 15 cm. Apply the inscribed square formula: s = r × √2 = 15 × 1.4142 = 21.213 cm.

Square area = s² = 21.213² = 450 sq cm (equivalently, 2r² = 2 × 225 = 450 sq cm).

Perimeter = 4 × 21.213 = 84.853 cm. Diagonal = 2r = 30 cm, confirming the diagonal spans the full diameter.

Square side = 21.213 cm | Square area = 450 sq cm | Fills 63.66% of disk

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Example 2 - Minimum Round Hole for a Square Peg (s = 20 mm)

An engineer needs the minimum circular hole diameter that will pass a square cross-section peg with side 20 mm.

Square side s = 20 mm. Circumscribed circle radius: r = s × √2 / 2 = 20 × 1.4142 / 2 = 14.142 mm.

Minimum hole diameter = 2r = 28.284 mm. Use a 29 mm or larger drill bit to allow clearance.

Circle area = π × 14.142² = 628.3 sq mm. Square area = 400 sq mm. Corner waste = 228.3 sq mm (36.34% of circle).

Circle radius = 14.142 mm | Minimum diameter = 28.284 mm | Circumference = 88.858 mm

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Example 3 - Square Courtyard in a Circular Garden (r = 25 m)

A landscape designer has a circular garden of radius 25 m and wants to inscribe a square paved courtyard.

Circle radius r = 25 m. Inscribed square side = 25 × √2 = 35.355 m.

Courtyard area = 2 × 625 = 1,250 sq m. Total garden area = π × 625 = 1,963.5 sq m.

The four curved corner segments = 1,963.5 − 1,250 = 713.5 sq m can be planted with greenery while the 1,250 sq m square is paved.

Courtyard side = 35.355 m | Courtyard area = 1,250 sq m | Paved area = 63.66% of garden

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Example 4 - Circular Log to Square Timber (s = 150 mm)

A sawmill wants a 150 mm by 150 mm square timber beam. Find the minimum circular log diameter needed.

Square side s = 150 mm. Circumscribed radius r = 150 × √2 / 2 = 106.07 mm.

Minimum log diameter = 2r = 212.13 mm. A 220 mm diameter log is the nearest standard size that works.

Material efficiency: beam area = 22,500 sq mm; log cross-section = π × 106.07² = 35,343 sq mm. Timber yield = 63.66%.

Minimum log radius = 106.07 mm | Log diameter = 212.13 mm | Timber yield = 63.66%

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❓ Frequently Asked Questions

What is the formula for a square inscribed in a circle?+

For a square inscribed in a circle of radius r, the side length is s = r times sqrt(2). This follows from the diagonal of the inscribed square equaling the diameter: diagonal = 2r. Since the diagonal of a square with side s is s times sqrt(2), solving gives s = 2r / sqrt(2) = r sqrt(2). Square area = 2r squared. Perimeter = 4r sqrt(2).

How do you find the radius of a circle circumscribed around a square?+

For a square with side s, the circumscribed circle radius is r = s sqrt(2) / 2. The diagonal of the square is s sqrt(2), and this diagonal is the diameter of the circumscribed circle. So the radius is half the diagonal: r = s sqrt(2) / 2. For a square with side 6: r = 6 times 1.4142 / 2 = 4.243 units. Diameter = 8.485 units.

What percentage of a circle does an inscribed square fill?+

An inscribed square always fills exactly 2 / pi = 63.66% of the circle area, regardless of circle size. Circle area = pi r squared. Square area = 2r squared. Ratio = 2 / pi = 0.6366 = 63.66%. The remaining 36.34% consists of four identical circular segments in the corners. This constant ratio is a well-known result of Euclidean geometry.

Why does the diagonal of the inscribed square equal the circle diameter?+

The diagonal of the inscribed square connects two opposite corners, both of which lie on the circle. The line connecting two points on a circle and passing through the center is a diameter. Since the diagonals of a square bisect each other at the center, each diagonal passes through the center of the circumscribed circle. Therefore every diagonal of an inscribed square is a diameter of the circle.

What is the side length of a square inscribed in a unit circle?+

For a unit circle (radius = 1), the inscribed square has side = 1 times sqrt(2) = 1.4142 units, area = 2 square units, perimeter = 4 sqrt(2) = 5.657 units, and diagonal = 2 units. The circle area is pi = 3.1416 square units, so the square fills 2/pi = 63.66% of the unit circle.

How is the square in a circle used in engineering?+

Engineers use the square-in-circle relationship in three common contexts: (1) Square peg in a round hole: the minimum circular hole for a square peg of side s requires diameter s sqrt(2). (2) Milling square timber from a circular log: a log of radius r yields a square beam of side r sqrt(2). (3) Cross-section optimization: inscribing a square shaft in a circular tube maximizes the shaft's cross-sectional area for a given tube diameter.

What is the area of a square inscribed in a circle of radius 5?+

For r = 5: side = 5 times sqrt(2) = 7.071 units; area = 2 times 25 = 50 square units; perimeter = 4 times 7.071 = 28.284 units; diagonal = 10 units (equals diameter). The circle area is pi times 25 = 78.54 square units. The square covers 50 / 78.54 = 63.66% of the circle.

What is the difference between an inscribed and a circumscribed square?+

An inscribed square fits inside a circle with all four corners on the circle. A circumscribed square surrounds a circle with the circle touching the midpoint of each side. For a circle of radius r: inscribed square side = r sqrt(2), circumscribed square side = 2r. The circumscribed square area (4r squared) is exactly twice the inscribed square area (2r squared). The circle fits between them with area pi r squared.

How do you construct a square inscribed in a circle?+

To construct an inscribed square with compass and straightedge: (1) Draw a circle with center O. (2) Draw a diameter AB. (3) Draw the perpendicular diameter CD through O. (4) Connect A to C, C to B, B to D, and D to A. The four points A, B, C, D form a square inscribed in the circle. This is one of the classic Euclidean constructions, known since antiquity.

Can you fit a larger square inside a circle than the inscribed one?+

No. The inscribed square (with corners on the circle) is the largest possible square that fits inside the circle. Any rotation of the square still has the same side length (rotation does not change dimensions), and any attempt to make the square larger would push corners outside the circle. The inscribed square with side r sqrt(2) is the unique largest square for a circle of radius r.

What is the circumscribed circle of a square with side 12?+

For a square with side 12: circumscribed radius r = 12 times sqrt(2) / 2 = 8.485 units; diameter = 16.971 units; circle area = pi times 72 = 226.19 square units; circumference = 2 pi times 8.485 = 53.31 units. The square area is 144 square units, filling 144 / 226.19 = 63.66% of the circle.

What is the perimeter of a square inscribed in a circle of radius 7?+

For r = 7: side = 7 times sqrt(2) = 9.899 units; perimeter = 4 times 9.899 = 39.598 units. Alternatively, perimeter = 4r sqrt(2) = 28 sqrt(2) = 39.598 units. Square area = 2 times 49 = 98 sq units. Diagonal = 14 units, equal to the circle diameter.

🔗 Related Calculators

📌 Quick Tips

💡The area of a square inscribed in a circle is always exactly 2r squared, where r is the radius. This comes directly from side = r times sqrt(2), so area = 2r squared.

💡The ratio of a square area to the area of its circumscribed circle is always 2 divided by pi, which equals approximately 63.66%. This never changes regardless of size.

💡For a unit circle (r = 1), the inscribed square has side length sqrt(2) = 1.4142 units, perimeter 4 times sqrt(2) = 5.657 units, and area = 2 square units.

💡The diagonal of the inscribed square always equals the diameter of the circle. This is because the diagonal spans the full diameter of the circle from one vertex to the opposite vertex.

💡To fit a square peg in a round hole, the hole diameter must be at least side times sqrt(2). For a 10 mm square peg, the minimum circular hole diameter is 14.14 mm.