### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

# Inscribed in a Circle

##### Stage: 3 Challenge Level:

Imagine extending the radius so that you have a horizontal diameter.

The hexagon is now split into two identical trapeziums (trapezia?).
Area of one trapezium $= {1\over 2}$ height $\times$ sum of parallel sides $$= {1\over 2}\ \times\ \sqrt0.75\ \times\ (2 + 1)$$ $$= {3\over 4} x \sqrt 3$$

The area of the hexagon is therefore: ${3\over2}\ \times\ \sqrt {3}$

The area of the triangle is half the area of the hexagon. triangle drawn in hexagon by joining alternate vertices

The area of the triangle is therefore ${3\over4}\ \times\ \sqrt {3}$
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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