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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?


A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

A Scale for the Solar System

Age 14 to 16
Challenge Level

Sun Venus Earth

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

When we measure something we use a scale : we consider the size of one thing in terms of another.

Make a fist with your hand (it's about the size of a large orange isn't it?), and then locate a point about 10m away from you.

If your fist was the Sun, the Earth would be more than 10 metres away and less than 1 mm in diameter (the tip of a ball-point pen).

Venus orbits the Sun three times in roughly two Earth years. On rare occasions Venus can be seen (from Earth) passing across the face of the Sun, this is called the Transit of Venus.

Usually Venus appears to pass either above or below the Sun because the plane of Venus' orbit is slightly tilted to the Earth's own orbit around the Sun. The Transit of Venus has only been observed and recorded six times since telescopes became available early in the 17th century.

The Transit of Venus last happened in 2004 and 2012, and will not happen again until 2117.

Although scientists can measure distances across space using radio waves, that's a recent ability, before that the only measurement that could be made was angle (the change in direction of view through a telescope) and time.

The Transit of Venus was a valuable observation because it provided the data with which the Earth to Sun distance could be calculated. This distance is called the Astronomical Unit and is used like a scale for the Solar System.

But even without a Transit of Venus to provide data astronomers could know the ratio between the Venus to Sun distance and the Earth to Sun distance. What could they observe and what calculation would they need to do?

Google will give you all sorts of interesting things connected with the Astronomical Unit and the Transit of Venus, but before you do that imagine you are living in the 17th century and don't have the internet, how might this Venus to Sun : Earth to Sun ratio be known?