Arc Radius Calculator (Width & Height)
Calculate the exact radius of curvature (R = (W² + 4H²) / 8H) from chord span and sagitta height, subtended central angle (θ), arc length (L), and parent circle geometry.
Quick Answer: How to Calculate the Radius of an Arch from Width & Height
To calculate the radius of a circular arch from its width (span) and height (rise), use the chord theorem formula: R = (W² + 4H²) / (8H), where W is the total horizontal span and H is the vertical rise measured from the springline to the highest point of the crown. For example, for an arched opening 36 inches wide with a 12-inch rise: square the width (36² = 1,296), add four times the square of the height (4 × 144 = 576), sum them (1,872), and divide by eight times the height (8 × 12 = 96). The resulting radius is 19.50 inches.
Interactive Sagitta & Circle Center Simulator
Drag the sagitta height slider to observe how flattening or deepening the arc dramatically changes the distance to the circle's center point ($R - H$) and total radius.
Offset-Coordinate "Mark-Out" Table & Interactive Plotter
For large arches where the swing radius exceeds your compass or room length. Simply draw your baseline, measure out horizontal intervals from center (X), and mark vertical height (Y) to plot perfect curves directly on wood or wall.
| Pt # | Dist from Center (X) | Height (Y) | Fraction | Drop |
|---|
Live Plywood Scribing Simulator
🔨 Carpentry Technique: Tap finish brads at coordinate marks. Bend a flexible batten against the pins to scribe a smooth curve.
Mathematical Derivation of the Radius Formula
The exact mathematical relationship between chord width $W$, sagitta height $H$, and radius $R$.
This elegant proof holds for every circular segment from shallow 1-degree curves to 180-degree semicircles.
Pivot drop $d$ is the distance to measure downward from the chord line to place your compass pivot pin.
Step-by-Step Worked Numeric Example
Suppose you measure a curved arch opening with a chord width of W = 48.0 inches and a sagitta height of H = 12.0 inches.
Common Arc Geometry & Trigonometry Calculation Errors
Prevent severe dimensional errors when converting radians, solving sagittas, or computing curved arc segments.
Mixing Degrees and Radians in Arc Length Formulas
Using angle in degrees directly in $L = R imes heta$ without converting to radians, producing an arc length 57 times larger than reality.
Always multiply degree angles by $pi / 180$ before computing arc length or sector dimensions, or use our direct degree formula $L = pi imes R imes heta / 180$.
Assuming Crown Rise Equals Circle Radius
Assuming rise ($H$) equals radius ($R$) for shallow segmental arches. This is only true for a full 180° semicircular arch ($H = W/2$).
Always calculate radius using the intersecting chord formula $R = (W^2 + 4H^2) / (8H)$ whenever rise is less than half the horizontal opening width.
Premature Decimal Rounding on Coordinate Offsets
Rounding intermediate radius or angle values to 1 decimal place, resulting in cumulative layout errors exceeding 1/2" at the arch shoulders.
Carry calculations to at least 4 decimal places internally before converting final jobsite mark-out measurements to 1/16" carpenter fractions.
Ordering Materials by Chord Width Instead of Arc Length
Measuring straight horizontal chord span ($W$) to order flexible jamb liners or trim, leaving the installer significantly short on material.
Calculate true Intrados Arc Length ($L$) along the curved soffit perimeter and add 10% for end trimming and miter joints.
Field Master Pro-Tips for Clean Execution
Related Geometric Solvers & Math
Explore complementary specialized geometry tools and layout solvers.
Geometric Solvers & Math
Tier 3 Math (5)- Arc Radius (from Width & Height)→
Solve radius R from chord width and sagitta height
- Arc Length & Perimeter Solver→
Calculate curved perimeter distance from radius and angle
- Arch Central Angle Calculator→
Compute subtended sweep angle in degrees and radians
- Sagitta (Arch Rise) Calculator→
Calculate vertical rise height from span and radius
- Complete Arc Solver (Any 2 Knowns)→
Enter any 2 parameters to solve all 5 dimensions
Frequently Asked Questions
Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.
Q1 How do you calculate radius from arc length and rise height (without knowing span)?
When chord span is unknown, radius R can be approximated using the Taylor expansion: R ≈ L² / (8H) + H / 6, where L is arc length and H is rise height. For exact engineering layout, our calculator solves R numerically from the transcendental equation L = 2R × arccos(1 − H / R).
Q2 What is the formula to calculate the radius of an arc from width and height?
The exact formula derived from the Pythagorean theorem is: Radius R = (W² + 4H²) / (8H), or equivalently R = W² / (8H) + H / 2, where W is the chord span/width and H is the rise height (sagitta). For example, for span W = 48" and height H = 12": R = (48² + 4×12²) / (8×12) = (2304 + 576) / 96 = 2880 / 96 = 30.00 inches.
Q3 How do you derive the arc radius formula?
Consider a right triangle formed by half the chord (W/2), the distance from chord to center (R - H), and the hypotenuse radius (R). By the Pythagorean theorem: R² = (W/2)² + (R - H)². Expanding: R² = W²/4 + R² - 2RH + H². Subtracting R² from both sides: 2RH = W²/4 + H². Dividing by 2H yields: R = W²/(8H) + H/2.
Q4 What is the sagitta of an arc?
The sagitta (from the Latin word for 'arrow') is the vertical height of an arc measured perpendicularly from the midpoint of the chord line to the highest point (crown apex) of the circular curve.
Q5 Can this formula calculate the radius of any circular arc?
Yes, this formula is universally valid for all circular arcs with sweep angles θ ≤ 180°. For a full 180° semicircle where H = W/2: R = (W² + 4(W/2)²) / (8(W/2)) = 2W² / 4W = W/2.