Clinical Medicine & Physiology
Body Roundness Index (BRI) Calculator: Thomas Waist-Height Ellipse Eccentricity Formula
Compute Diana M. Thomas et al.’s (2013) geometric Body Roundness Index BRI = 364.2 - 365.5 * SQRT(1 - ((WC / (2 * PI()))^2) / ((0.5 * Height)^2)) and sagittal ellipse eccentricity e from Waist and Height.
Model torso geometry as a prolate ellipse to quantify visceral adiposity and U-shaped mortality risk!
Formulated by mathematician Diana M. Thomas et al. (2013, Obesity) using 3D optical body-scanner geometry, the Body Roundness Index (BRI) models the human silhouette as an upright ellipse with semi-major vertical axis a = 0.5 * Height and horizontal waist radius b = WC / (2 * PI())! As abdominal visceral fat expands waist circumference relative to stature, ellipse eccentricity e = SQRT(1 - (b / a)^2) = SQRT(1 - ((WC / (2 * PI())) / (0.5 * Height))^2) drops from ~0.99 (slender vertical ellipse) toward 0 (perfect sphere)!
Mapping eccentricity linearly onto a 1-to-20 clinical scale gives BRI = 364.2 - 365.5 * e = 364.2 - 365.5 * SQRT(1 - ((WC / (2 * PI()))^2) / ((0.5 * Height)^2))! Large NHANES all-cause mortality studies (Zhang et al., 2024, JAMA Netw. Open) highlight healthy mid-range BRI (~3.0 to 5.5) flanked by elevated risk at frailty (< 3.4) and severe central visceral obesity (>= 6.9)!
| Anthropometric Profile (Height = 175 cm) | Torso Eccentricity e | Body Roundness Index (BRI) | Geometric / Clinical Tier |
|---|---|---|---|
| Slender Adult (Waist = 76 cm) | 0.9904 | 2.21 | Lean Reference Silhouette |
| Moderate Central Gain (Waist = 92 cm) | 0.9859 | 3.85 | Mid-Range Silhouette |
| Severe Central Obesity (Waist = 108 cm) | 0.9805 | 5.82 | Elevated Central Adiposity |
Thomas (2013) Body Roundness Index (BRI)
Evaluates 364.2 - 365.5 * SQRT(1 - ((WC / (2 * PI()))^2) / ((0.5 * Height)^2)) for Height B2 (cm) and Waist B3 (cm, requiring WC / (2 * PI()) < 0.5 * Height).
=IF(OR(B2 < 100, B2 > 240, B3 < 40, B3 >= B2 * PI()), "Invalid Height or Waist Circumference", ROUND(364.2 - 365.5 * SQRT(1 - (((B3 / (2 * PI())) ^ 2) / ((0.5 * B2) ^ 2))), 2))Torso Sagittal Ellipse Eccentricity e (0 < e < 1)
Computes dimensionless geometric eccentricity SQRT(1 - ((WC / (2 * PI())) / (0.5 * Height))^2).
=IF(OR(B2 < 100, B2 > 240, B3 < 40, B3 >= B2 * PI()), "Invalid Height or Waist Circumference", ROUND(SQRT(1 - (((B3 / (2 * PI())) ^ 2) / ((0.5 * B2) ^ 2))), 4))Try the example
Use the original values or change them to test your understanding.