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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Of All the Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

In this problem, students are encouraged to measure areas using a triangular unit and come up with a general formula for finding the area of a tilted triangle. It brings together geometrical thinking, algebra, and sequences, and offers a different perspective on Pythagoras' Theorem.

Possible approach

This printable worksheet may be useful: Of All The Areas

For a similar problem using the more familiar context of a square dotty grid and leading to Pythagoras's Theorem, see Tilted Squares. For introductory work on using a triangular unit to measure area, see Isometric Areas and More Isometric Areas.
You may wish to print off isometric paper, or use the interactive dotty grid environment.

Without speaking, draw equilateral triangles in the usual orientation on the board, starting as shown in the problem. Write the areas of the first two or three triangles and place question marks next to the rest.

Allow time for reflection and discussion, drawing out ideas such as the use of non-standard units and the interesting result of square numbers.

Present the idea of tilted triangles, discussing how this might be defined before setting the challenge posed in the second part of the problem. A good point to discuss is how we know the triangles are equilateral - those who are convinced that the triangles are equilateral could explain their reasoning to those who aren't.

It is worthwhile giving the class some time to draw the diagrams and try to come up with their own methods for finding the areas of the tilted triangles, and to share the methods that they find, but if they are struggling to find an efficient way, the pictures in Getting Started might be useful as prompts.
Once a few areas have been found, encourage the students to make conjectures about the areas of much larger triangles with a tilt of 1, and to justify their ideas. The lesson could be structured in a similar way to the lesson in these videos of the task Tilted Squares.

Key Questions

How do you know the tilted triangles are equilateral?
How can you find the area of a tilted triangle in terms of the unit equilateral triangle?
Can you find a generalisation for the area of a tilted triangle?

Possible Extension

Can you find a general rule for finding the area of any sized triangle with any tilt?
Are there any areas that it's impossible to make with a tilted triangle?

Possible Support

You could start with the problems Isometric Areas and More Isometric Areas.

Focus on the justification that the tilted triangles are equilateral, and on calculating the areas rather than seeking generalisations.