There are 74 results
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 11 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?
Where to Land
Age 14 to 16
Challenge Level
Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?
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?
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?
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?
Kite in a Square
Age 14 to 18
Challenge Level
Can you make sense of the three methods to work out what fraction of the total area is shaded?
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?
Babylon Numbers
Age 11 to 18
Challenge Level
Can you make a hypothesis to explain these ancient numbers?
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 ?
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.
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.
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. . . .
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.
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.
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 16
Challenge 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 16
Challenge Level
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
Cutting a Cube
Age 11 to 14
Challenge Level
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
Take a Square
Age 14 to 16
Challenge 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 16
Challenge 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 16
Challenge Level
Can you find a relationship between the area of the crescents and the area of the triangle?
Tilting Triangles
Age 14 to 16
Challenge 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?
Are You Kidding
Age 14 to 16
Challenge Level
If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?
Matter of Scale
Age 14 to 16
Challenge Level
Can you prove Pythagoras' Theorem using enlargements and scale factors?
Rhombus in Rectangle
Age 14 to 16
Challenge Level
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
The Medieval Octagon
Age 14 to 16
Challenge Level
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Holly
Age 14 to 16
Challenge Level
The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.
NRICH is part of the family of activities in the Millennium Mathematics Project.