task(H5): polygon area calculation - shoelace polygonArea, formatArea with quadratic scaling, measure-area tool, properties panel area display

frontend/src/components/PropertiesPanel.tsx

@@ -1,4 +1,6 @@
 import React, { useState, useEffect } from 'react';
+import { polygonArea } from '../tools/modification/geometry';
+import { formatArea } from '../utils/format';
 import type { PropertiesPanelProps } from '../types/ui.types';
 import { formatInputValue, parseDistance, type UnitType } from '../utils/format';
 
@@ -69,7 +71,13 @@ const PropertiesPanel: React.FC<PropertiesPanelProps> = ({
     <>
       <div className="panel-section">
         <div className="panel-section-title">
-          <span>Auswahl {el ? `· ${el.type}` : ''}</span>
+          <span>Auswahl {el ? `· ${el.type}
+        {((el.type === 'polygon' || el.type === 'polyline') && ((el.properties.points as Array<{ x: number; y: number }> | undefined)?.length ?? 0) >= 3) && (
+          <div className="prop-row">
+            <label className="prop-label">Fläche</label>
+            <span className="prop-value">{formatArea(polygonArea(el.properties.points as Array<{ x: number; y: number }>), unit, scaleFactor)}</span>
+          </div>
+        )}` : ''}</span>
           <div style={{ display: 'flex', gap: '4px' }}>
             {onDelete && el && (
               <button