Enter radius, diameter, circumference, or area and get every circle measurement: inscribed square, hexagon, triangle sides, circumscribed square, area ratios, and more.
This all-in-one circle measurements calculator goes beyond the basic radius-diameter-circumference-area set. Enter any one known property and the tool computes every important circle dimension, including the side lengths of inscribed and circumscribed regular polygons.
The inscribed square — the largest square that fits inside a circle — has side s = r√2, where r is the radius. Its area is exactly 2r², which is always 2/π ≈ 63.66% of the circle's area. The circumscribed square — the smallest square that contains the entire circle — has side equal to the diameter, and its area is d² = 4r². The circle fills π/4 ≈ 78.54% of the circumscribed square — this is the classic "squaring the circle" ratio.
For regular hexagons, the inscribed hexagon has side length equal to the radius (a beautiful geometric fact), and its area is (3√3/2)r² ≈ 2.598r². The inscribed equilateral triangle has side r√3 and area (3√3/4)r².
These relationships are essential in engineering, packing problems, material cutting, and design. How much material is wasted when cutting circles from square sheets? What size circle do you need to inscribe a given square? This calculator answers all such questions with a single input. Use the visual comparison bars to see how the different shapes relate in size, and the reference table to look up values for common radii.
This circle measurements calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Enter, Unit and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Basic: d = 2r, C = 2πr, A = πr² Inscribed square side: s = r√2, area = 2r² Circumscribed square side: s = d = 2r, area = 4r² Inscribed regular hexagon side: s = r, area = (3√3/2)r² Inscribed equilateral triangle side: s = r√3, area = (3√3/4)r² Circle/circumscribed square area ratio: π/4 ≈ 0.7854 Circle/inscribed square area ratio: π/2 ≈ 1.5708
Result: For mode=radius, val=12.13, unit=mm, the tool returns the solved circle measurements outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circle measurements formulas and reports derived values, checks, and classifications automatically.
This page is tailored to circle measurements, with outputs tied directly to the form fields (Enter, Unit). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
The inscribed square with side s = r√2 ≈ 1.414r, where r is the circle's radius. Its diagonal equals the circle's diameter.
The circumscribed square with side equal to the diameter (2r). The circle fits perfectly inside, touching all four sides.
A regular hexagon inscribed in a circle of radius r consists of 6 equilateral triangles. Each triangle has two sides equal to r (radii) and an angle of 60° between them, making the third side also r.
The waste is 1 − π/4 ≈ 21.46%. A circle uses about 78.54% of its circumscribed square's area.
For a circle of radius r, the inscribed equilateral triangle has side s = r√3 ≈ 1.732r and area (3√3/4)r².
Yes. Select "Area" as the input mode, enter the value, and the calculator derives the radius (r = √(A/π)) and all other measurements.