GMY-02

Heron area

Any triangle.

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GeometryHeron

Governing equation

s=(a+b+c)/2,\\;\\Delta=\\sqrt{s(s-a)(s-b)(s-c)}

$a \in \left[0.5, 20\right]$

where

a: a (—)
b: b (—)
c: c (—)
\Delta: Δ (—)

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.

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①

Governing law — red

s=(a+b+c)/2,\\;\\Delta=\\sqrt{s(s-a)(s-b)(s-c)}

$a \in \left[0.5, 20\right]$

$a = 0.5$ $\Delta = 0$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.5,\quad a_{\max} = 20

$a \in \left[0.5, 20\right]$

$a = 0.5$ $a = 0.5$

Along the curve

③

Response curve — green

s=(a+b+c)/2,\\;\\Delta=\\sqrt{s(s-a)(s-b)(s-c)}

$a \in \left[0.5, 20\right]$

$a = 0.5$ $\Delta = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta}{\mathrm{d}a}

$a \in \left[0.5, 20\right]$

$a = 0.623$ $\Delta^{\prime} = 0$

Along the curve

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-02 — Heron area) is the form associated with Heron. Working symbols: $a$ , $b$ , $c$ $\rightarrow$ $\Delta$ . Any triangle.

Purpose

Purpose: compute

\Delta

from

a

b

c

in Geometry via

s=(a+b+c)/2,\\;\\Delta=\\sqrt{s(s-a)(s-b)(s-c)}

Any triangle. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 5.000\,\mathrm{—}

b = 6.000\,\mathrm{—}

c = 7.000\,\mathrm{—}

, the governing relation

s=(a+b+c)/2,\\;\\Delta=\\sqrt{s(s-a)(s-b)(s-c)}

yields

\Delta = 14.6969\,\mathrm{—}

. Any triangle. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a5 —b b6 —c c7 —

Outputs

Δ \Delta14.6969 —

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Narration of this film

Any triangle.

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