# Angles, Polygons and Geometrical Proof - Stage 3 ### Square It

##### Age 11 to 16 Challenge Level:

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. ### Square Coordinates

##### Age 11 to 14 Challenge Level:

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? ### Triangles in Circles

##### Age 11 to 14 Challenge Level:

Can you find triangles on a 9-point circle? Can you work out their angles? ### Subtended Angles

##### Age 11 to 14 Challenge Level:

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it? ### Right Angles

##### Age 11 to 14 Challenge Level:

Can you make a right-angled triangle on this peg-board by joining up three points round the edge? ##### Age 11 to 14 Challenge Level:

A game for 2 or more people, based on the traditional card game Rummy. ### Shapely Pairs

##### Age 11 to 14 Challenge Level:

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow... ### Property Chart

##### Age 11 to 14 Challenge Level:

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid? ### Semi-regular Tessellations

##### Age 11 to 16 Challenge Level:

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations? ##### Age 11 to 16 Challenge Level:

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem? ### Which Solids Can We Make?

##### Age 11 to 14 Challenge Level:

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids? ### Opposite Vertices

##### Age 11 to 14 Challenge Level:

Can you recreate squares and rhombuses if you are only given a side or a diagonal? ##### Age 11 to 14 Challenge Level:

How many questions do you need to identify my quadrilateral? ##### Age 11 to 14 Challenge Level:

We started drawing some quadrilaterals - can you complete them? ### Of All the Areas

##### Age 14 to 16 Challenge Level:

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid? ### Estimating Angles

##### Age 7 to 14 Challenge Level:

How good are you at estimating angles? ### Tilted Squares

##### Age 11 to 14 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted? ### Hidden Squares

##### Age 11 to 14 Challenge Level:

Can you find the squares hidden on these coordinate grids? ##### Age 11 to 14 Challenge Level: ### Polygon Rings

##### Age 11 to 14 Challenge Level:

Join pentagons together edge to edge. Will they form a ring? ### Star Polygons

##### Age 11 to 14 Challenge Level:

Draw some stars and measure the angles at their points. Can you find and prove a result about their sum? ### Polygon Pictures

##### Age 11 to 14 Challenge Level:

Can you work out how these polygon pictures were drawn, and use that to figure out their angles? ### An Equilateral Triangular Problem

##### Age 11 to 14 Challenge Level:

Take an equilateral triangle and cut it into smaller pieces. What can you do with them? ### Triangle in the Corner

##### Age 11 to 14 Short Challenge Level:

A triangle is shaded within a regular hexagon. Can you find its area? ### Funky Hexagon

##### Age 11 to 14 Short Challenge Level:

Can you find the value of x in this diagram? ### Perimeter in a Hexagon

##### Age 11 to 14 Short Challenge Level:

Can you find the perimeter of this triangle inscribed in a hexagon? ### Tricky Tessellations

##### Age 11 to 14 Short Challenge Level:

Can you work out the fraction of the tiles that are painted black in this pattern? ### Descending Angles

##### Age 11 to 14 Short Challenge Level:

Given four of the angles in two triangles, can you find the smallest angle overall? ### Clock Face Angles

##### Age 11 to 14 Short Challenge Level:

The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock? ### Overlapping Beer Mats

##### Age 11 to 14 Short Challenge Level:

Can you find the area of the overlap when these two beer mats are placed on top of each other? ### Clock Angle

##### Age 11 to 14 Short Challenge Level:

What is the angle between the the hands of a clock at 8:24? ### Two Triangles

##### Age 11 to 14 Short Challenge Level:

Prove that the angle marked $a$ is half the size of the angle marked $b$. ### Inner Rectangle

##### Age 11 to 14 Short Challenge Level:

If the shape on the inside is a rectangle, what can you say about the shape on the outside? ### No Rights

##### Age 11 to 14 Short Challenge Level:

Prove that the angle bisectors of a triangle can never meet at right angles. ### Perimeter Puzzle

##### Age 11 to 14 Short Challenge Level:

If four copies of this triangle are joined together to form a parallelogram, what is the largest possible perimeter of the parallelogram? ### Diagonal Side

##### Age 11 to 14 Short Challenge Level:

Can you work out the area of a square drawn on a diagonal? ##### Age 11 to 14 Short Challenge Level:

Find the sum of all of the angles denoted by letters in this diagram ### Angles Inside

##### Age 11 to 14 Challenge Level:

Draw some angles inside a rectangle. What do you notice? Can you prove it? ### Fraction of a Square

##### Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded? ### Square Within a Square Within...

##### Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded?