Quarter Circle Calculator
Enter one radius to calculate the chord, 90° arc, full perimeter, quarter-circle area, and the area left between the arc and its enclosing square.
Inputs
Live results
132.73 cm²
One fourth of the area of a full circle with the same radius.
18.38 cm
20.42 cm
46.42 cm
36.27 cm²
For a radius of 13 centimeters, the quarter area is 132.73 square centimeters and the perimeter is 46.42 centimeters.
Formula breakdown
c = r × √2
L = πr ÷ 2
P = 2r + L
A₁ = πr² ÷ 4
A₂ = r² – A₁
Calculation details
| Measurement | Formula | Value |
|---|---|---|
| Radius | r | 13 cm |
| Chord | r × √2 | 18.38 cm |
| Quarter arc | πr ÷ 2 | 20.42 cm |
| Perimeter | 2r + πr ÷ 2 | 46.42 cm |
| Quarter area | πr² ÷ 4 | 132.73 cm² |
| External area | r² – πr² ÷ 4 | 36.27 cm² |
Values are calculated with JavaScript's full-precision value of π and rounded to two decimal places only for display. The Excel workbook keeps the underlying numeric values.
How to use the Quarter Circle Calculator
What this calculator does
This calculator turns the radius of a 90-degree circular sector into five related measurements: Chord, Quarter arc, Perimeter, Quarter area, and External area. The external area is the portion of the radius-by-radius square that lies outside the curved quarter circle. The results are exact geometric identities before display rounding; they do not include material thickness, cutting kerf, measurement uncertainty, or construction allowances.
When to use it
Use it when laying out a rounded corner, estimating the surface area of a quarter-circle panel, checking the curved edge length of a 90-degree arc, or comparing the curved sector with its enclosing square. The formulas follow the standard circle relationships summarized in the OpenStax geometric formulas reference.
How to calculate
- The calculator opens with a ready-to-use demonstration: Radius = 13 and Length unit = centimeters. Its valid example workbook is available immediately through Download Excel.
- Replace the Radius with your measured or planned value. Enter a positive decimal, optionally with U.S.-style comma grouping, such as 2.75 or 1,250.5. Scientific notation and decimal-comma input are rejected to prevent silent reinterpretation.
- Select the Length unit. Changing from centimeters to millimeters, meters, inches, or feet converts the current radius rather than merely relabeling it.
- Read the live result cards and the Calculation details table. Linear results use the selected length unit; area results use its squared form.
- Select Download Excel to create a validated Office Open XML workbook containing current inputs, results, formulas, and typed numeric cells. Reset clears the demonstration and all calculated content. Download Excel then remains unavailable until a complete valid Radius is entered again.
Input guide
Radius is required. It accepts a positive plain number or a correctly grouped U.S.-style number up to 1,000,000,000,000. A realistic example is 13. Increasing Radius increases Chord, Quarter arc, and Perimeter in direct proportion, while Quarter area and External area grow with the square of the radius. Common mistakes include entering a diameter instead of a radius, typing a unit symbol into the numeric field, using a decimal comma such as 1,5, or entering zero.
Length unit is required and offers millimeters, centimeters, meters, inches, and feet. It controls the unit shown for Radius, Chord, Quarter arc, and Perimeter, while Quarter area and External area use the corresponding square unit. Unit conversion preserves the physical dimension. For example, 13 cm becomes 130 mm, not 13 mm.
Output guide
The header summary pills labeled Radius, Area, and Perimeter repeat the current radius, Quarter area, and Perimeter for quick scanning. Quarter area is the exact sector area πr²/4 and is the primary result. Chord is the straight line joining the arc endpoints, equal to r√2. Quarter arc is one fourth of a full circumference, equal to πr/2. Perimeter adds the arc to the two straight radii. External area equals r² minus the quarter-circle area and is always positive for a positive radius.
The Calculation details table uses the columns Measurement, Formula, and Value. It is a traceable summary rather than a separate estimate: every row is generated from the same current model as the result cards and Excel workbook. Zero or blank results indicate an incomplete or invalid input state, not a geometric solution.
Worked example
With Radius set to 13 cm, the chord is 13√2 = 18.3848 cm, displayed as 18.38 cm. The quarter arc is π × 13 ÷ 2 = 20.4204 cm, so the perimeter is 26 + 20.4204 = 46.42 cm. The quarter area is π × 13² ÷ 4 = 132.7323 cm², displayed as 132.73 cm². The enclosing square has area 169 cm², leaving an External area of 169 – 132.7323 = 36.27 cm².
How the geometry works
A quarter circle is a circular sector with a central angle of 90 degrees, or π/2 radians. Sector arc length and area scale with the fraction of the full 360-degree circle. That is why the arc is one fourth of 2πr and the area is one fourth of πr². A deeper derivation of sector relationships is available in the Wolfram MathWorld circular sector reference.
The chord crosses the two radius endpoints. Those endpoints and the center form a right isosceles triangle whose legs both equal r. Applying the Pythagorean theorem gives c² = r² + r², so c = r√2. The perimeter is not one fourth of the circumference alone: it includes the curved arc plus both radius edges.
Units, precision, and practical interpretation
Length and area are different dimensions. A result in centimeters becomes square centimeters only after two lengths are multiplied. NIST explains that area is measured in square units, including mm², cm², and m², in its guide to SI units of area. For metric conversions, the calculator uses exact powers of ten; for inches and feet, it uses exact international conversion factors through meters. NIST's SI units of length guide provides the underlying metric relationships.
Displayed values are rounded to two decimal places for readability, but calculations and exported numeric cells retain more precision. For fabrication or surveying, the practical accuracy is limited by the quality of the radius measurement. Measure from the circle center to the arc, not across the full width; across the full circle would be the diameter, which is twice the radius.