RTH-07

Equivalent square

s = 2ab/(a+b) = 4A/P. Sterling's equivalent square.

Reading speed

Beam dataSterling

Governing equation

s=\dfrac{2ab}{a+b}=\dfrac{4A}{P}

$a \in \left[2, 40\right]$

where

a: Side a (cm)
b: Side b (cm)
s: Equivalent square (cm)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

s=\dfrac{2ab}{a+b}=\dfrac{4A}{P}

$a \in \left[2, 40\right]$

$a = 2$ $s = 3.2$

Along the curve

②

Domain of the variable — blue

a_{\min} = 2,\quad a_{\max} = 40

$a \in \left[2, 40\right]$

$a = 2$ $a = 2$

Along the curve

③

Response curve — green

s=\dfrac{2ab}{a+b}=\dfrac{4A}{P}

$a \in \left[2, 40\right]$

$a = 2$ $s = 3.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}s}{\mathrm{d}a}

$a \in \left[2, 40\right]$

$a = 2.239$ $s^{\prime} = 1.222$

Along the curve

Lecture brief

Historical brief

The linear-quadratic model, BED, tissue-maximum ratio and Clarkson scatter turned radiotherapy into fraction arithmetic. MU calculation on these sheets is that clinical closed form. This sheet (RTH-07 — Equivalent square) is the form associated with Sterling. Working symbols: $a$ , $b$ $\rightarrow$ $s$ . A rectangle and a square of side s have nearly the same scatter if 4A/P matches.

Purpose

Purpose: compute

s

from

a

b

in Radiotherapy via

s=\dfrac{2ab}{a+b}=\dfrac{4A}{P}

s = 2ab/(a+b) = 4A/P. Sterling's equivalent square. Use it when a real radiotherapy question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 12.000\,\mathrm{cm}

b = 8.000\,\mathrm{cm}

, the governing relation

s=\dfrac{2ab}{a+b}=\dfrac{4A}{P}

yields

s = 9.60\,\mathrm{cm}

. A rectangle folding onto a square of side s. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Side a a12 cmSide b b8 cm

Outputs

Equivalent square s9.60 cm

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A rectangle folding onto a square of side s.

A rectangle and a square of side s have nearly the same scatter if 4A/P matches.

Reading speed

Watch on YouTube