# Resources tagged with: Pythagoras' theorem

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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem ### 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? ### Floored

##### Age 14 to 16Challenge 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? ### Holly

##### Age 14 to 16Challenge 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. ### Round and Round

##### Age 14 to 16Challenge Level

Prove that the shaded area of the semicircle is equal to the area of the inner circle. ### Circle Packing

##### Age 14 to 16Challenge Level

Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ... ### Two Circles

##### Age 14 to 16Challenge Level

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap? ### Square Pegs

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle? ### Some(?) of the Parts

##### Age 14 to 16Challenge 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 ### Hex

##### Age 11 to 14Challenge 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. ### The Pillar of Chios

##### Age 14 to 16Challenge Level

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle. ### A Chordingly

##### Age 11 to 14Challenge Level

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. ### Three Four Five

##### Age 14 to 16Challenge 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. ### Get Cross

##### Age 14 to 16Challenge 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? ##### 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. ### 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. ### Medallions

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

If a ball is rolled into the corner of a room how far is its centre from the corner? ### Squareo'scope Determines the Kind of Triangle

##### Age 11 to 14

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

##### Age 14 to 16Challenge Level

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . . ### 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? ### Circle Scaling

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it? ### 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. ### 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! ### Pythagorean Triples II

##### Age 11 to 16

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

##### Age 14 to 16Challenge 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? ### Are You Kidding

##### Age 14 to 16Challenge 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? ### All Is Number

##### Age 7 to 14 ### 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. ### Rhombus in Rectangle

##### Age 14 to 16Challenge 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. ### Babylon Numbers

##### Age 11 to 18Challenge Level

Can you make a hypothesis to explain these ancient numbers? ### 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. . . . ### Grid Lockout

##### Age 14 to 16Challenge Level

What remainders do you get when square numbers are divided by 4? ### 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 ?. ### 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. ### 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? ### 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? ### 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 ? ### 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 . ### Rectangular Pyramids

##### Age 14 to 18Challenge Level

Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges? ### Garden Shed

##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge 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. ### 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. ### The Medieval Octagon

##### Age 14 to 16Challenge Level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please. ### The Old Goats

##### Age 11 to 14Challenge 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. . . . ### 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?