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Broad Topics > Angles, Polygons, and Geometrical Proof > Angles - points, lines and parallel lines

### Olympic Turns

##### Stage: 2 Challenge Level:

This task looks at the different turns involved in different Olympic sports as a way of exploring the mathematics of turns and angles.

### Turning

##### Stage: 1 Challenge Level:

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

### Six Places to Visit

##### Stage: 2 Challenge Level:

Can you describe the journey to each of the six places on these maps? How would you turn at each junction?

### How Safe Are You?

##### Stage: 2 Challenge Level:

How much do you have to turn these dials by in order to unlock the safes?

### Round and Round and Round

##### Stage: 3 Challenge Level:

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

### Semi-regular Tessellations

##### Stage: 3 Challenge Level:

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

### Orbiting Billiard Balls

##### Stage: 4 Challenge Level:

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

### Octa-flower

##### Stage: 5 Challenge Level:

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

### Estimating Angles

##### Stage: 3 Challenge Level:

How good are you at estimating angles?

### Watch the Clock

##### Stage: 2 Challenge Level:

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

### Angles Inside

##### Stage: 3 Challenge Level:

Draw some angles inside a rectangle. What do you notice? Can you prove it?

### Robotic Rotations

##### Stage: 3 and 4 Challenge Level:

How did the the rotation robot make these patterns?

### Shogi Shapes

##### Stage: 3 Challenge Level:

Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring?

### Being Resilient - Primary Geometry

##### Stage: 1 and 2 Challenge Level:

Geometry problems at primary level that may require resilience.

### Being Resourceful - Primary Geometry

##### Stage: 1 and 2 Challenge Level:

Geometry problems at primary level that require careful consideration.

### Being Collaborative - Primary Geometry

##### Stage: 1 and 2 Challenge Level:

Geometry problems for primary learners to work on with others.

### Being Curious - Primary Geometry

##### Stage: 1 and 2 Challenge Level:

Geometry problems for inquiring primary learners.

### Same Length

##### Stage: 3 and 4 Challenge Level:

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

### Witch's Hat

##### Stage: 3 and 4 Challenge Level:

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

### Cylinder Cutting

##### Stage: 2 and 3 Challenge Level:

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

### Which Solids Can We Make?

##### Stage: 3 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?

##### Stage: 2, 3 and 4 Challenge Level:

A metal puzzle which led to some mathematical questions.

### Interacting with the Geometry of the Circle

##### Stage: 1, 2, 3 and 4

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

### Dotty Circle

##### Stage: 2 Challenge Level:

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

### Flight Path

##### Stage: 5 Challenge Level:

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

### Overlapping Squares

##### Stage: 2 Challenge Level:

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

### Spiroflowers

##### Stage: 5 Challenge Level:

Analyse these repeating patterns. Decide on the conditions for a periodic pattern to occur and when the pattern extends to infinity.

### Spirostars

##### Stage: 5 Challenge Level:

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

### Making Maths: Clinometer

##### Stage: 3 Challenge Level:

Make a clinometer and use it to help you estimate the heights of tall objects.

### Making Maths: Equilateral Triangle Folding

##### Stage: 2 and 3 Challenge Level:

Make an equilateral triangle by folding paper and use it to make patterns of your own.

### Tessellating Hexagons

##### Stage: 3 Challenge Level:

Which hexagons tessellate?

### Three Tears

##### Stage: 4 Challenge Level:

Construct this design using only compasses

##### Stage: 4 Challenge Level:

Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?

### Angle Measurement: an Opportunity for Equity

##### Stage: 3 and 4

Suggestions for worthwhile mathematical activity on the subject of angle measurement for all pupils.

### Watch Those Wheels

##### Stage: 1 Challenge Level:

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

### Sweeping Hands

##### Stage: 2 Challenge Level:

Use your knowledge of angles to work out how many degrees the hour and minute hands of a clock travel through in different amounts of time.

### Right Angle Challenge

##### Stage: 1 Challenge Level:

How many right angles can you make using two sticks?

### Pythagoras

##### Stage: 2 and 3

Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.

### Maurits Cornelius Escher

##### Stage: 2 and 3

Have you ever noticed how mathematical ideas are often used in patterns that we see all around us? This article describes the life of Escher who was a passionate believer that maths and art can be. . . .

### Flower

##### Stage: 5 Challenge Level:

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

### Lunar Angles

##### Stage: 5 Challenge Level:

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

### LOGO Challenge 7 - More Stars and Squares

##### Stage: 3 and 4 Challenge Level:

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

### LOGO Challenge 8 - Rhombi

##### Stage: 2, 3 and 4 Challenge Level:

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

### LOGO Challenge 1 - Star Square

##### Stage: 2, 3 and 4 Challenge Level:

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

### Take the Right Angle

##### Stage: 2 Challenge Level:

How many times in twelve hours do the hands of a clock form a right angle? Use the interactivity to check your answers.

### Hand Swap

##### Stage: 4 Challenge Level:

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

### Right Time

##### Stage: 3 Challenge Level:

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

### On Time

##### Stage: 3 Challenge Level:

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

### A Problem of Time

##### Stage: 4 Challenge Level:

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

### Square World

##### Stage: 5 Challenge Level:

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?