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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem ### Garden Shed

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

Can you minimise the amount of wood needed to build the roof of my garden shed? ### Pythagoras Proofs

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

Can you make sense of these three proofs of Pythagoras' Theorem? ### Partly Circles

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

What is the same and what is different about these circle questions? What connections can you make? ### Nicely Similar

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

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle? ### Where Is the Dot?

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

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height? ### Cubestick

##### Age 16 to 18 Challenge Level:

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise. ### Where to Land

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

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible? ### The Spider and the Fly

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

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly? ### Far Horizon

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

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see? ### 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? ### Inscribed in a Circle

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

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius? ### Belt

##### Age 16 to 18 Challenge Level:

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r. ### Pythagoras for a Tetrahedron

##### Age 16 to 18 Challenge Level:

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . . ### Semi-detached

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

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius. ### Hex

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

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other. ### Zig Zag

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

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line? ### Napkin

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

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed . ### Compare Areas

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

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle? ### Three Cubes

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

Can you work out the dimensions of the three cubes? ### Orthogonal Circle

##### Age 16 to 18 Challenge Level:

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally. ### Ladder and Cube

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

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground? ### Logosquares

##### Age 16 to 18 Challenge Level:

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares. ### Baby Circle

##### Age 16 to 18 Challenge Level:

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle? ### Kite in a Square

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

Can you make sense of the three methods to work out the area of the kite in the square? ### Generating Triples

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more? ### The Fire-fighter's Car Keys

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

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?. ### Under the Ribbon

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

A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ? ### Spherical Triangles on Very Big Spheres

##### Age 16 to 18 Challenge Level:

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere. ### Liethagoras' Theorem

##### Age 7 to 14

Liethagoras, Pythagoras' cousin (!), was jealous of Pythagoras and came up with his own theorem. Read this article to find out why other mathematicians laughed at him. ### Pythagoras

##### Age 7 to 14

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. ### Golden Construction

##### Age 16 to 18 Challenge Level:

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio. ### The Dangerous Ratio

##### Age 11 to 14

This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying. ### Classic Cube

##### Age 16 to 18 Challenge Level:

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card? ### Slippage

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

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . . ### All Is Number

##### Age 7 to 14

Read all about Pythagoras' mathematical discoveries in this article written for students. ### All Tied Up

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

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be? ### Circle Scaling

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

Describe how to construct three circles which have areas in the ratio 1:2:3. ### Circle Box

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

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit? ### Xtra

##### Age 14 to 18 Challenge Level:

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations. ### Squ-areas

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

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . . ### Pythagoras Mod 5

##### Age 16 to 18 Challenge Level:

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5. ### Chord

##### Age 16 to 18 Challenge Level:

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles. ### The Dodecahedron

##### Age 16 to 18 Challenge Level:

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron? ### Corridors

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

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner. ### Circumnavigation

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

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle. ### About Pythagorean Golden Means

##### Age 16 to 18

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio? ### Incircles Explained

##### Age 16 to 18

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers. ### Squareo'scope Determines the Kind of Triangle

##### Age 11 to 14

A description of some experiments in which you can make discoveries about triangles. ### Picturing Pythagorean Triples

##### Age 14 to 18

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself. ### Pythagorean Triples II

##### Age 11 to 16

This is the second article on right-angled triangles whose edge lengths are whole numbers.