What is the formula for SAG?

The geometric formula for SAG (Sagitta) is S = R - sqrt(R^2 - (d/2)^2), where R is the radius and d is the diameter or chord length.

Why is SAG important in suspension?

Proper SAG ensures the suspension stays in the correct part of its travel, providing better traction and preventing the vehicle from bottoming out.

How do you calculate SAG for a lens?

For lenses, the SAG formula is used to determine the thickness of the lens based on the radius of curvature of the glass surface.

SAG Calculator - Free & Accurate Online Calculator

📐 SAG Calculator

Precision SAG Calculator: Master Curvature and Suspension Geometry

Primary Goal	Input Metrics	Output Results	Why Use This?
Measure Arc Depth	Radius ($R$) & Chord/Diameter ($d$)	Sagitta ($S$)	Essential for suspension tuning, lens manufacturing, and architectural arch integrity.

Understanding SAG (Sagitta)

In geometry, SAG (technically known as the sagitta) represents the vertical distance between the center of a circular arc and its chord. It measures the “drop” or depth of a curve. In mechanical contexts, such as vehicle suspension, SAG refers to the amount the suspension compresses under a specific load, which is a specialized application of the same geometric principle.

Calculating SAG is vital for ensuring that curved structures can bear intended loads and that mechanical systems operate within their optimal “stroke” range. Without accurate SAG data, arches may lack structural stability, and suspension systems may suffer from poor traction or frequent bottoming out.

Who is this for?

Mechanical Engineers: Tuning mountain bike or motorcycle suspension for rider weight.
Architects: Calculating the rise of structural arches and domes.
Opticians: Determining the thickness and curvature of optical lenses.

The Logic Vault

The geometric SAG formula is derived from the Pythagorean theorem applied to a circle’s radius and its chord.

$$S = R – \sqrt{R^2 – \left(\frac{d}{2}\right)^2}$$

Variable Breakdown

Name	Symbol	Unit	Description
Sagitta (SAG)	$S$	$m / in$	The depth or “drop” of the curve.
Radius	$R$	$m / in$	The distance from the center of the circle to the arc.
Chord (Diameter)	$d$	$m / in$	The straight-line distance between the arc’s endpoints.

Step-by-Step Interactive Example

Let’s calculate the SAG for an architectural arch where the Radius ($R$) is 10 m and the span or Diameter ($d$) is 13 m.

Identify Inputs:$R = \mathbf{10}$$d = \mathbf{13}$
Calculate the Half-Chord ($\frac{d}{2}$):$$\frac{13}{2} = \mathbf{6.5}$$
Apply the Pythagorean Component:$$\sqrt{10^2 – 6.5^2} = \sqrt{100 – 42.25} = \sqrt{57.75} \approx \mathbf{7.5993}$$
Final Subtraction:$$10 – 7.5993 = \mathbf{2.4007 \text{ m}}$$

Information Gain: The “Static vs. Dynamic” Variable

In suspension engineering, users often confuse Static SAG with Dynamic SAG. Static SAG is the compression under the vehicle’s own weight, while Dynamic (or Rider) SAG includes the operator.

Expert Edge: For optimal performance, your suspension SAG should typically fall between $25\%$ and $33\%$ of the total travel. If your geometric calculation shows a SAG outside this window, you likely need to adjust the Preload or change the Spring Rate, rather than just adjusting the dampening.

Strategic Insight by Shahzad Raja

“After 14 years of architecting technical SEO tools, I’ve found that the biggest pitfall in SAG calculation is Unit Inconsistency. When using the Sagitta formula, $R$ and $d$ must be in the same unit. Furthermore, if you are using this for optical lenses or high-precision engineering, ensure your $R$ is the ‘Radius of Curvature’ and not the ‘Aperture Radius,’ as mixing these up results in a 50% error margin in your depth calculation.”

Frequently Asked Questions

How do I calculate SAG for a suspension?

Measure the distance between the axle and a fixed point on the frame while the bike is on a stand (unloaded), then subtract the distance measured while the rider is sitting on the bike (loaded).

What is the SAG for a 5m radius and 10m diameter?

Using the formula:

$$5 – \sqrt{5^2 – (10/2)^2} = 5 – \sqrt{25 – 25} = \mathbf{5 \text{ m}}$$

(This represents a perfect semi-circle).

Does rider weight change the SAG?

Yes. In mechanical suspension, SAG is directly proportional to the force applied ($F = kx$). Heavier riders will require more preload to maintain the same SAG.

Related Tools

Diameter Calculator: Determine the chord length of your arc for more complex geometries.
Spring Rate Calculator: Find the correct spring stiffness based on your desired SAG.
Unicode Tools (Category): Access specialized engineering symbols like $\varnothing$ (diameter), $\approx$ (approximation), and $\perp$ (perpendicular) for your technical drawings.