Lots to find out, lots to explore. This task looks at the different turns involved in different Olympic sports as a way of exploring the mathematics of turns and angles. This square piece of paper has been folded and creased. Prove Pythagoras’ Theorem for right-angled spherical triangles. Be patient this problem may be slow to load. Can you work out the length of its base?

Dotty Circle Age 7 to 11 Challenge Level: That is, they should all experience challenging, intellectually worthwhile and coherently-structured mathematics. Shogi tiles can form interesting shapes and patterns How do I know? What fraction of the square is shaded? Watch this film carefully. A spiropath is a sequence of connected line segments end to end taking different directions.

Can you find its area?

## Angle Measurement: an Opportunity for Equity

A quadrilateral changes shape with the edge lengths constant. 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. Prove that with the given ratios of the radii the petals touch and fit perfectly. Investigate how this is possible.

Weekly Problme 23 – A triangle has been drawn inside this circle. Weekly Problem 16 – Can you figure out how far the robot has travelled by the time it is first facing due East? Olympic Turns Age 7 to 11 Challenge Level: Of course, for some pupils, the task could become one of proving why the angles are as they are, rather than doing any measuring.

Among these might be. Alison made a conjecture: All pupils have a right to enrichment.

# Estimating Angles :

Can you prove it in more than one way? Find the exact values of some trig.

porblem Weekly Problem 26 – The diagram shows two equilateral triangles. Right Angles Age 11 to 14 Challenge Level: How high is the wall that this cat is lying on? Link Puzzle Age 7 to 16 Challenge Level: Can you find the area of the overlap when these two beer mats are placed on top of each other?

Dotty Circle Age 7 to 11 Challenge Level: To support problme 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. What is the sum of the interior angles of the resulting polygon?

Can you work out the means of these distributions using numerical methods?

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Quadrilaterals Age 7 to 11 Challenge Level: A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. Construct two equilateral triangles on a straight line. What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? Making Sixty Age 14 to 16 Challenge Level: Weekly Problem 4 – In the figure given in the problem, calculate the length of an edge.

A similar task could be repeated for different point-circles – those with eight or twelve points are relatively straightforward to construct. Orthogonal Circle Age 16 to 18 Challenge Level: The exercise itself was repeated twice more over the page. How long will it be before they are at right angles again?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Have a good look at these images.