# Resources tagged with: Pythagoras' theorem

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There are 81 NRICH Mathematical resources connected to Pythagoras' theorem, you may find related items under Pythagoras and Trigonometry.

Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem

### Pythagoras Proofs

##### Age 14 to 16Challenge Level

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

### Partly Circles

##### Age 14 to 16Challenge Level

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

### Nicely Similar

##### Age 14 to 16Challenge 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 16Challenge 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 18Challenge 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 16Challenge 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 16Challenge 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 16Challenge 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?

### Inscribed in a Circle

##### Age 14 to 16Challenge 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 18Challenge 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 18Challenge 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 16Challenge 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.

### Zig Zag

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

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

### Three Cubes

##### Age 14 to 16Challenge Level

Can you work out the dimensions of the three cubes?

### Orthogonal Circle

##### Age 16 to 18Challenge Level

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

##### Age 14 to 16Challenge 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 18Challenge 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 18Challenge 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 16Challenge 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 16Challenge 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?

### Babylon Numbers

##### Age 11 to 18Challenge Level

Can you make a hypothesis to explain these ancient numbers?

### The Fire-fighter's Car Keys

##### Age 14 to 16Challenge 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 16Challenge 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 18Challenge 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.

### Golden Construction

##### Age 16 to 18Challenge Level

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

### Classic Cube

##### Age 16 to 18Challenge 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 16Challenge 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 Tied Up

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

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

### Circle Box

##### Age 14 to 16Challenge 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 18Challenge 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 16Challenge 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 18Challenge 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 18Challenge 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 18Challenge 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 16Challenge 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.

##### Age 14 to 16Challenge Level

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

##### 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?

### 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.

### Pythagorean Triples I

##### Age 11 to 16

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

### Squaring the Circle and Circling the Square

##### Age 14 to 16Challenge Level

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

### In a Spin

##### Age 14 to 16Challenge Level

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

### Take a Square

##### Age 14 to 16Challenge Level

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

### Semi-square

##### Age 14 to 16Challenge Level

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

### Crescents and Triangles

##### Age 14 to 16Challenge Level

Can you find a relationship between the area of the crescents and the area of the triangle?

### Tilting Triangles

##### Age 14 to 16Challenge Level

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?