# Shape: Level 6

The key idea of shape at level 6 is that abstract geometric properties may be used to solve real problems.

There are two basic aspects of this thread: the use of angles and similar triangles to make practical measurements and the use of angles in circles. The former has more immediate applications in the real world.

Underlying each of the several achievement objectives in this thread, there are a number of more restricted ideas. These are the importance of right angled triangles in measurement; the use of deductive power in mathematics (through proving things about circles); a three dimensional problem may be reduced to a sequence of two dimensional problems; angles that subtended by the same arc are equal (the properties of isosceles triangles have interesting and useful consequences); and related triangles and quadrilaterals inscribed in circles have interesting properties

It should be noted that the similar triangles work of this thread is an extension of proportion in number.

This key idea develops from both the key idea of shape at level 5 and the key idea of transformation at level 5. The shape work looks at angle properties of circles (instead of angles properties of lines and polygons) and the transformation material is extended to more work on trigonometric ratios and by taking the applications of Pythagoras’ Theorem into three dimensions.

Shape has important connections at Level 6 to measurement (especially in the discussion of accuracy) as well as to Levels 7 and 8 and even to tertiary mathematics. This can be seen immediately in patterns and relationships level 7, where the trigonometry is extended to the sine and cosine rules and into both two and three dimensional situations. But the trigonometric functions are also widely used in all of calculus (at school and beyond) both for their own importance as functions that describe real situations and as a tool to integrate other functions. Similar shapes and Pythagoras’ Theorem are also tools that can be helpful in later mathematics.