# Resources tagged with: Pythagoras' theorem

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##### Other tags that relate to Painted Purple
Cubes & cuboids. Real world. Pythagoras' theorem. Nets. Coordinates - 3D. Interactivities. Optimisation. Rotations. Games. Visualising.

### There are 75 results

Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem ### 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? ### Weighty Problem

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

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . . ### 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? ### 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? ### 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? ### Trice

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

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR? ### Three Cubes

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

Can you work out the dimensions of the three cubes? ### 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. ### The Old Goats

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

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . . ### 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? ### Star Gazing

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

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star. ### Matter of Scale

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

Prove Pythagoras' Theorem using enlargements and scale factors. ### All Is Number

##### Age 7 to 14 ### 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? ### 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? ### 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. ### 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. ### Some(?) of the Parts

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

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle ### 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. ### Square Pegs

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

Which is a better fit, a square peg in a round hole or a round peg in a square hole? ### Floored

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

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded? ### 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. ### Garden Shed

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

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

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

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing? ### 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. ### 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. ### 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. . . . ### 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 ?. ### 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? ### Medallions

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

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one? ### Pythagorean Triples

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

How many right-angled triangles are there with sides that are all integers less than 100 units? ### Tennis

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

A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area? ### Squareo'scope Determines the Kind of Triangle

##### Age 11 to 14

A description of some experiments in which you can make discoveries about triangles. ### Isosceles

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

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas. ### 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. ### A Chordingly

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

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. ### 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? ### 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? ### 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? ### Grid Lockout

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

What remainders do you get when square numbers are divided by 4? ### 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? ### Six Discs

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

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases? ### Pythagoras Proofs

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

Can you make sense of these three proofs of Pythagoras' Theorem? ### 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. ### 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? ### 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 ? ### 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? ### 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? ### Three Four Five

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

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.