Calculate inscribed square in circle dimensions (\(a = r\sqrt{2} = \frac{D}{\sqrt{2}}\), \(A = 2r^2\), \(63.66\%\) area coverage) and circumscribed circle in square dimensions (\(r = \frac{a}{2}\), \(A = \frac{\pi a^2}{4}\), \(78.54\%\) area coverage). Features log timber milling yield, off-cut scrap ratios, and multi-unit conversions.
Select configuration and input any known circle or square parameter.
Square Area: 200.00 cm² • Circle Area: 314.16 cm² • Coverage: 63.66%
| 📏 Circle Radius (\(r\)) | 10.000 cm |
| 📏 Circle Diameter (\(D\)) | 20.000 cm |
| 🔄 Circle Circumference (\(C\)) | 62.832 cm |
| 🔵 Circle Area (\(A_{\text{circle}}\)) | 314.159 cm² |
| ⬛ Square Side Length (\(a\)) | 14.142 cm |
| 📐 Square Diagonal (\(d\)) | 20.000 cm |
| 📏 Square Perimeter (\(P\)) | 56.569 cm |
| ⬛ Square Area (\(A_{\text{square}}\)) | 200.000 cm² |
| 📊 Area Coverage Ratio | 63.662% (2/π) |
| 🪵 Off-Cut / Scrap Waste | 114.159 cm² (36.338%) |
| 🍕 Single Corner Segment | 28.540 cm² |
The geometric relationship between concentric squares and circles is central to sawmill timber processing, semiconductor silicon wafer fabrication, CNC sheet metal punching, and architectural dome design. Our Square in a Circle Calculator solves critical mathematical and industrial problems:
Calculates the largest possible square timber post (\(a = \frac{D}{\sqrt{2}}\)) that can be milled from a round tree log of diameter \(D\), while accurately tracking the \(36.34\%\) off-cut wood waste.
Determines the maximum square integrated circuit (IC) or sensor die footprint that fits onto circular 200mm, 300mm, or 450mm semiconductor silicon wafers.
Calculates the exact scrap percentage (\(21.46\%\)) when punching circular blanks from square sheet metal sheets (\(A_{\text{circle}} = \frac{\pi}{4}A_{\text{square}}\)).
Solves all 8 geometric parameters (radius, diameter, circumference, circle area, side length, diagonal, perimeter, square area) from any single known input.
Select Square in a Circle (Inscribed Square) or Circle in a Square (Inscribed Circle).
Choose from Radius (\(r\)), Diameter (\(D\)), Side Length (\(a\)), Perimeter, or Area.
Input your measurement with custom unit dropdowns (m, cm, mm, in, ft, yd).
Review side lengths, area coverage (\(63.66\%\) or \(78.54\%\)), scrap area, and copy the report.
$$a = r\sqrt{2} = \frac{D}{\sqrt{2}}, \quad d_{\text{square}} = 2r = D$$ $$A_{\text{square}} = a^2 = 2r^2 = \frac{D^2}{2}, \quad A_{\text{circle}} = \pi r^2$$ $$\text{Coverage Ratio} = \frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi} \approx \mathbf{63.662\%}$$ $$A_{\text{offcut}} = (\pi - 2)r^2 \approx \mathbf{36.338\%}$$
$$r = \frac{a}{2}, \quad D = a, \quad d_{\text{square}} = a\sqrt{2}$$ $$A_{\text{circle}} = \frac{\pi a^2}{4}, \quad A_{\text{square}} = a^2$$ $$\text{Coverage Ratio} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4} \approx \mathbf{78.5398\%}$$ $$A_{\text{corners}} = \left(1 - \frac{\pi}{4}\right)a^2 \approx \mathbf{21.4602\%}$$
A round tree log has a usable diameter of \(D = 30\,\text{cm}\). What is the maximum side length of a square timber post that can be milled from it, and what is the off-cut waste?
Solution: \(a = \frac{30}{\sqrt{2}} \approx \mathbf{21.213\,\text{cm}}\). Square Area = \(a^2 = \mathbf{450.00\,\text{cm}^2}\). Log Area = \(\frac{\pi \times 30^2}{4} \approx \mathbf{706.86\,\text{cm}^2}\). Off-cut scrap = \(706.86 - 450 = \mathbf{256.86\,\text{cm}^2}\) (\(36.34\%\)).
A square sheet metal plate measures \(a = 50\,\text{cm}\) on each side. A circular disc of maximum diameter is stamped out. Find the circle area and scrap metal percentage.
Solution: \(r = \frac{50}{2} = 25\,\text{cm}\). Circle Area = \(\pi \times 25^2 \approx \mathbf{1,963.50\,\text{cm}^2}\). Square Area = \(50^2 = \mathbf{2,500.00\,\text{cm}^2}\). Scrap area = \(2,500 - 1,963.50 = \mathbf{536.50\,\text{cm}^2}\) (\(21.46\%\)).
A circle has radius \(r = 10\,\text{cm}\). Compare the area of the circumscribed square to the inscribed square.
Solution: Inscribed Square Area = \(2r^2 = 2(10^2) = \mathbf{200.00\,\text{cm}^2}\). Circumscribed Square Area = \((2r)^2 = 4(10^2) = \mathbf{400.00\,\text{cm}^2}\). The circumscribed square has exactly double (\(2\times\)) the area of the inscribed square!
Standard dimensions from unit circles to industrial tree logs and CNC sheet blanks.
| Configuration | Circle Diameter (\(D\)) | Square Side (\(a\)) | Circle Area | Square Area | Coverage Ratio |
|---|---|---|---|---|---|
| Unit Circle Inscribed Sq | \(2.000\) | \(1.414\) (\(\sqrt{2}\)) | \(3.142\) (\(\pi\)) | \(2.000\) | \(63.66\%\) |
| Log Milling (\(D = 30\,\text{cm}\)) | \(30.000\,\text{cm}\) | \(21.213\,\text{cm}\) | \(706.858\,\text{cm}^2\) | \(450.000\,\text{cm}^2\) | \(63.66\%\) |
| 10-Inch Wafer Die | \(10.000\,\text{in}\) | \(7.071\,\text{in}\) | \(78.540\,\text{in}^2\) | \(50.000\,\text{in}^2\) | \(63.66\%\) |
| Circle in 10-Inch Box | \(10.000\,\text{in}\) | \(10.000\,\text{in}\) | \(78.540\,\text{in}^2\) | \(100.000\,\text{in}^2\) | \(78.54\%\) |
| CNC Sheet Blank (\(a = 1\,\text{m}\)) | \(1.000\,\text{m}\) | \(1.000\,\text{m}\) | \(0.785\,\text{m}^2\) | \(1.000\,\text{m}^2\) | \(78.54\%\) |
Authoritative answers to common questions about inscribed squares in circles, circumscribed circles in squares, area ratios, and timber log milling.
