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To draw a golden rectangle, start with a square, bisect one side, then use a compass to extend its length. The finished rectangle’s long side divided by its short side is approximately 1.618, the golden ratio. You can divide that rectangle into smaller golden rectangles and add quarter-circle arcs for the familiar hand-drawn golden-ratio spiral.
What you need
- A ruler or straightedge
- A drawing compass
- A sharp pencil
- Optionally, a set square or another right-angled aid
A ruler and compasses let you transfer lengths geometrically; a measured drawing is quicker but depends on accurate ruler marks. Clear right angles and careful marking matter because small errors become more noticeable in the smaller shapes. See the NRICH construction activity and the New Zealand Curriculum’s golden spirals resource.
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Draw a golden rectangle with a compass and straightedge
The construction below begins with a square. NRICH uses a 10 cm square, but any convenient side length works.
- Draw a square ABCD. Label the vertices in order around the square, and use AB as the side you will extend.
- Find the midpoint of AB. Mark it M, so AM and MB are equal.
- Set the compass to the opposite corner. Place its point at M and open it to C. With the compass centered at M, draw an arc that crosses the line AB extended beyond B. Label the crossing point E.
- Complete the rectangle. Use the ruler and a right-angled aid to draw the perpendicular and parallel sides needed to form a rectangle with the square’s original height and the extended length AE.
- Check the proportion. Measure the rectangle’s long and short sides and divide the longer measurement by the shorter. A hand-drawn construction should give a result close to 1.618.
The compass transfer works because MC is the hypotenuse of a right triangle whose legs are half the square’s side and the full side. That transferred length sets the rectangle’s extended dimension. For a classroom diagram and construction instructions, see NRICH’s Golden Construction. If you prefer, you can measure and mark the extended length with a ruler instead, accepting the small errors that measurement may introduce.
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Why the rectangle has the golden ratio
The golden ratio is φ = (1 + √5)/2, approximately 1.62. A golden rectangle has long side divided by short side equal to this proportion; the Smithsonian National Museum of American History uses the same definition in its Golden Rectangle collection entry.
In the construction, the new length is the hypotenuse of a right triangle with legs one-half the square’s side and one full side. Completing the rectangle with that length and the original square’s height produces the golden proportion. Hand-drawn measurements will usually be close rather than mathematically exact.
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Turn the rectangle into a spiral approximation
Repeatedly remove a square from the remaining golden rectangle. Each leftover rectangle has the same proportions, so the process creates a sequence of smaller rectangles. Quarter-circle arcs drawn inside the squares make the familiar spiral-like design.
- Mark the first square. Within the golden rectangle, identify a square whose side equals the rectangle’s shorter side.
- Keep the leftover rectangle. The area beside that square is a smaller golden rectangle. Rotate the direction in which you mark the next square.
- Repeat inward. In each leftover rectangle, mark off a square with sides equal to that rectangle’s short side. Continue while the remaining spaces are large enough to draw clearly.
- Add the arcs. Draw a quarter-circle arc within each square, connecting its endpoints to the arc in the next square. Work inward through the nested squares to form the spiral-like curve.
This is the usual geometric quarter-circle approximation, not an exact logarithmic golden spiral. The nested rectangles and successive arcs are described by NRICH’s Golden Mathematics and the Glencoe/McGraw-Hill Golden Rectangles handout.
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Keep the drawing accurate
- Use a sharp pencil and make small, clear marks for the midpoint and arc intersections.
- Keep the compass opening fixed while transferring a length, and hold the point steady at the center.
- Check that each new square has equal sides and that the remaining rectangle’s edges meet at right angles.
- Expect later, smaller rectangles to show the effect of any earlier marking errors; the New Zealand Curriculum resource notes that these errors can accumulate.
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