KEVOS® MAKEOS · Geometry
Four-Arc Oval Approximating an Ellipse (End and Side Radii and Their Centres)
To lay out an oval of a given length and width from four circular arcs that approximates an ellipse, what end radius, side radius and arc centres should I use, and how close is its area to the true ellipse?
Four-arc oval construction from the handbook (Brueckner method), which still works when the length is four or more times the width: end radius r = (B² ÷ 2A)(A ÷ B)^0.38, the distance X of the side-arc centre beyond the oval centre X = (A²/4 − Ar + Br − B²/4) ÷ (B − 2r) and the side radius R = B/2 + X, with the arc centres for layout, the angle of each arc, the tangent points, and the perimeter and area of the oval against the true ellipse. A smaller or larger end radius can be entered to make the oval thinner or fatter.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Major axis A (overall length) | mm | Overall length of the oval along its major axis. |
| Minor axis B (overall width) | mm | Overall width of the oval across its minor axis. |
| End radius r | choice | Use the handbook’s equation (1) for the end radius, or enter your own radius: smaller makes the oval thinner at the ends, larger makes it fatter. |
| End radius r (when entered) | mm | End radius used when you choose to enter it; it must be less than half the width. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Major axis A (overall length)
- What it is
- Overall length of the oval measured along its major axis.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer oval has larger side arcs and a smaller end radius for the width.
- Get it right
- Use the finished outside or inside size the part needs.
What this means: Minor axis B (overall width)
- What it is
- Overall width of the oval measured across its minor axis.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the end radius and the side-arc radius.
- Get it right
- Must be less than the length.
What this means: End radius r
- What it is
- Whether the end radius comes from the handbook equation or is entered.
- Units and format
- Choose one of 2 options from the list: From the handbook equation (1); Entered (thinner or fatter oval).
- Example
- The form opens with “From the handbook equation (1)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the end radius and with it every other arc.
- Get it right
- Start with the equation; enter a radius to make the ends sharper or fuller.
What this means: End radius r (when entered)
- What it is
- End radius of the oval used when you choose to enter it yourself.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger end radius moves the side-arc centres closer to the oval.
- Get it right
- Must be less than half the width; smaller gives a thinner oval.
Method and formulas
- r = (B² ÷ 2A)(A ÷ B)^0.38 (handbook equation 1), or the radius you enter.
- X = (A²/4 − Ar + Br − B²/4) ÷ (B − 2r) (equation 2); R = B/2 + X (equation 3).
- End-arc centres lie on the major axis at ±(A/2 − r) from the oval centre; side-arc centres lie on the minor axis at X beyond the centre, on the side away from the arc they draw.
- The arcs meet on the line joining their centres, at a half-angle α = atan((A/2 − r) ÷ X) from the minor axis; each side arc spans 2α and each end arc 180° − 2α.
- Perimeter 4αR + 2(π − 2α)r; area 2R²α + r²(π − 2α) − 2(A/2 − r)X; ellipse area πAB ÷ 4.
- Example: a 400 × 100 mm oval uses end arcs of radius 21.169 mm and side arcs of radius 590.2 mm with centres 540.2 mm beyond the oval centre.
Limitations
- The oval is made of circular arcs that meet tangentially; it is close to, but not exactly, an ellipse.
- The major axis must be longer than the minor axis, and the end radius less than half the width.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Lay out an oval flange, duct, tank or opening with compasses or arcs in CAD
- Program an oval as four arcs on a CNC machine
- Compare an arc oval with the true ellipse area
Who it is for
- Sheet metal workers
- Drafters
- CNC programmers
- Pattern makers
What you get
- End radius r and side radius R
- Distance of the side-arc centres from the oval centre
- Centres of the four arcs for layout
- Angle of each arc and the tangent points
- Perimeter and area of the oval and the ellipse area
Interpreting the result
- The area difference from the true ellipse shows how close the approximation is; a larger end radius makes the oval fuller at the ends and a smaller one makes it thinner.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 51. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Constructing a four-centre oval
In Oval Sheet Metal Work by the Numbers
What it covers: Find a four-centre oval's side radius, centres and tangent points from its length, width and end radius, choose the end radius, mark it out with a trammel, and catch a radius worked from full sizes.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Four-Arc Oval Approximating an Ellipse (End and Side Radii and Their Centres) calculator