Resources tagged with: Pythagoras' theorem

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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem

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.

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?

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?

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.

Two Circles

Age 14 to 16 Challenge 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?

Circle Packing

Age 14 to 16 Challenge 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? ...

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?

Fitting In

Age 14 to 16 Challenge 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. . . .

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.

Floored

Age 14 to 16 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?

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

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?

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?

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.

The Pillar of Chios

Age 14 to 16 Challenge 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.

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?

Round and Round

Age 14 to 16 Challenge Level:

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

Medallions

Age 14 to 16 Challenge Level:

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

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.

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.

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.

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 .

Circle Scaling

Age 14 to 16 Challenge Level:

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

Pythagoras Proofs

Age 14 to 16 Challenge Level:

Can you make sense of these three proofs of 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?

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?

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?

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?

Squareo'scope Determines the Kind of Triangle

Age 11 to 14

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

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.

Ball Packing

Age 14 to 16 Challenge Level:

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

Pythagorean Triples II

Age 11 to 16

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

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?

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!

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.

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?

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?

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 ?.

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?

All Is Number

Age 7 to 14

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

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.

Grid Lockout

Age 14 to 16 Challenge Level:

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

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?

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.

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. . . .

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?

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.

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 ?

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.

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?