Quarter Circle Calculator - Chord, Arc, Perimeter, Area

A quarter circle calculator is a tool that takes a single radius (or the diameter of the parent circle) and returns every defining parameter of the shape: the chord, the curved arc, the full perimeter, the inside area, and the external corner area, with the inside area also shown in its in-terms-of-pi form. The shape itself is the region bounded by two perpendicular radii and the arc between their endpoints, which is exactly the slice of a 90 degree pie.

• • Classroom geometry: Check radius, chord, arc, perimeter, and area for homework and lesson examples.
• • Rounded corners on plans: Estimate the area of a rounded interior corner, quarter-round trim, or a patio with a curved cutout.
• • Food and serving portions: Estimate the surface area of a 90 degree pie slice, quarter pizza, or quarter-pan portion for portion math.
• • Square insets and external corners: Compute the external area left over when the shape is cut out of a square corner.

This shape is one fourth of a full circle, with a 90 degree central angle and the two straight sides meeting at a right angle. The perimeter adds the two straight sides on top of the arc, unlike the full circle.

In every parameter the shape is half of a semicircle. The inside area is pi r squared over 4 instead of pi r squared over 2, the arc is pi r over 2 instead of pi r.

For the full circle of the same radius, the Circle Calculator covers the area, circumference, radius, and diameter in one place.

The calculator picks the radius, evaluates the five formulas, and shows the inside area in the (r^2 / 4) pi form and decimal.

• r: radius of the shape, the distance from the center vertex to either end of the arc
• d: diameter of the parent circle, equal to 2 * r
• Chord: straight-line distance between the two arc endpoints, equal to r * sqrt(2)
• Arc: curved boundary of the slice, equal to pi * r / 2
• Perimeter: full boundary, equal to the arc plus the two straight radii
• Area: inside area, equal to pi * r^2 / 4
• External Area: r^2 minus the inside area, the corner of the enclosing square that the curve does not cover

The in-terms-of-pi display is the (r^2 / 4) pi answer before rounding. Keep it for textbook problems and exact checks; use the decimal for material lists and design dimensions.

According to Wolfram MathWorld - Circular Sector, a circular sector of central angle theta in radians has area one half r squared theta and arc length r theta, so this slice at theta equals pi over 2 has area pi r squared over 4

According to Wikipedia - Quarter circle, the shape subtends a right angle at the center, so its area and arc length are exactly one fourth of the full circle's area and circumference

When the slice is half a circle instead of a quarter, the Semicircle Area Calculator returns the area and the diameter-driven boundary for a 180 degree sector.

These four ideas decide whether the formula you are using matches the shape you are actually measuring.

A common source of error is comparing this shape to a semicircle. A semicircle has a central angle of 180 degrees, so its area is pi r squared over 2 and its arc is pi r. This shape is half of a semicircle, with the area and the arc halved again. The chord is the straight diagonal between the two arc endpoints, equal to r times sqrt 2, while the arc is pi r over 2. For r = 5 the chord is 7.07 and the arc is 7.85, a difference of 0.78 that matters for trim and edging work.

When the slice is not exactly a quarter circle, the Arc Length Calculator solves for the curved arc length at any central angle.

Pick the input method that matches the measurement you already have, then read the five result rows in order.

1. 1 Pick the input method: Choose Use Radius when you have the radius, or Use Diameter when the drawing only labels the full width of the parent circle.
2. 2 Enter the radius or the diameter: Type the radius in the radius field, or type the diameter in the diameter field. The other field is ignored for the chosen method.
3. 3 Read the chord and the arc: Use the chord for the straight diagonal distance across the slice, and the curved arc for the boundary, equal to pi r over 2.
4. 4 Read the perimeter: Use the perimeter for the total boundary length, which is the curved arc plus the two straight radii.
5. 5 Read the area and the external area: Use the inside area for material counts, paint coverage, or inside comparisons. Use the external area for the leftover corner of the enclosing r by r square.
6. 6 Cross-check with the diameter method: Switch to Use Diameter and enter the diameter for the same shape. The chord, arc, perimeter, and area should match the radius run to two decimals.

A designer is laying out a patio with a curved seating nook. With radius = 8 feet the method returns chord = 11.31, arc = 12.57, perimeter = 28.57, area = 50.27, and external area = 13.73. The arc gives the edging length, the area gives the paver count, and the external area shows how much of the r by r square stays outside the curve.

When the design needs the inside area of the full circle in square feet, the Square Footage Circle Calculator covers that working-unit case in one step.

A calculator that returns all five parameters from a single radius is easier to use than a chain of separate tools.

• • Single-radius entry: Type one radius and the calculator returns chord, arc, perimeter, area, and external area together.
• • Two input methods: Use radius for the most direct calculation, or diameter when only the full width is labeled.
• • In-terms-of-pi display: The exact symbolic form (r^2 / 4) pi is shown alongside the decimal for hand checks and exact algebra.
• • External area included: The output is the leftover corner of the r by r square, useful for trim math, inset cutouts, and corner tile pieces.
• • Decimal friendly: Decimal values work for measured sketches, scaled drawings, and design dimensions.
• • Unit consistency: Chord, arc, and perimeter share the linear unit of the radius; area outputs share the matching square unit.

The single-radius entry is the most useful entry point for design and classroom work. The diameter method is a convenience for measured objects like a round tabletop where the full width is easier to read.

When only the diameter is on hand and the radius is not, the Circle Diameter Calculator recovers the radius from the measured full width before any other step.

The calculator runs on compact math, but a few measurement decisions affect whether the answer matches the real shape.

• • This calculator does not solve for the shape from a chord or a different central angle. A non-90-degree slice is a sector, with area theta over 360 times pi r squared for theta in degrees.
• • Results are geometric estimates. Real material takeoffs may need allowances for seams, overlap, cutting waste, or coating, especially for curved trim and tile.
• • Rounded output can differ by a few hundredths from a hand calculation that rounds after each intermediate step. The internal computation keeps full precision before the display rounds to two decimals.

The five factors and three limitations above cover the cases that come up most often in classroom and design work. For non-right-angle slices, use the general sector formulas with the central angle in radians or degrees.

According to Math is Fun - Circle Sector, the area of a sector is theta over 360 times pi r squared and the arc length is theta over 180 times pi r for a central angle theta in degrees

When the central angle is not 90 degrees, the Central Angle Calculator converts between degrees, radians, arc length, and radius for the more general sector formulas.