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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. . . .

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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?

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

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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?

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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?

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

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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.

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

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.

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

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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?

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

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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?

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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?

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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? ...

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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.

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A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?

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

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

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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.

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

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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. . . .

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

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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?

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.

Challenge Level

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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

Read all about Pythagoras' mathematical discoveries in this article written for students.

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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?

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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. . . .

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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?

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

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It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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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?

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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?

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

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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?

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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.

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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.

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

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If a ball is rolled into the corner of a room how far is its centre from the corner?

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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?

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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. . . .

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

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

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

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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.

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Prove Pythagoras' Theorem using enlargements and scale factors.

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