Plane geometry and geometric constructions applied to design
This domain opens the Madrid PAU paper with a construction exercise: a proportion, a regular polygon, an oval or a graphic scale. The exam tests compass-and-straightedge reasoning, so the mark depends on visible construction lines (arcs left in place, labelled points, clean intersections) and not on a figure measured with the ruler.
Core rules
- Thales / proportionality: to divide AB into n equal parts, draw a ray from A, step off n equal compass units, join the last point to B and draw parallels through the other marks.
- Media proporcional: in a right triangle the altitude on the hypotenuse satisfies h² = m·n; alternatively use the semicircle on the sum of the two segments.
- Golden section: locate C on AB with AB/AC = AC/CB = φ ≈ 1.618; the golden rectangle and modular layouts are built from it.
- Regular polygons given the side: general method - raise the perpendicular at one end, strike an arc of radius equal to the side, divide it into n parts and take the second division as the circumcentre.
- Ovals and ovoides: four-centre constructions; every tangency point lies on the line joining the two centres involved.
Worked example: golden section of AB = 50 mm
Raise the perpendicular at B, mark D with BD = 25 mm (half the segment), and draw AD = √(50² + 25²) = 55.9 mm. With centre D and radius 25 mm, cut AD at E: AE = 55.9 − 25 = 30.9 mm. With centre A and radius AE, cut AB, giving AC = 30.9 mm. Check: 30.9 × 1.618 = 50 mm, so C divides AB in the golden ratio; the same radius gives the side of the regular decagon inscribed in that radius.
Traps that cost marks
Erasing construction arcs (they are evidence of the method); confusing media proporcional with tercera/cuarta proporcional; constructing a polygon on the side but measuring the radius with the ruler; leaving the tangency points of an oval unmarked; drawing the golden rectangle from the diagonal instead of from the section; rounding to the millimetre in a construction that is exact by geometry.
In design questions the same tools reappear (grids, logos, typographic modules), so always name the geometric property that justifies the result.