Search by Topic

Resources tagged with Pythagoras' theorem similar to Hex:

Filter by: Content type:
Stage:
Challenge level: Challenge Level:1 Challenge Level:2 Challenge Level:3

There are 70 results

Broad Topics > 2D Geometry, Shape and Space > Pythagoras' theorem

problem icon

Hex

Stage: 3 Challenge Level: Challenge Level:1

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.

problem icon

Pythagorean Triples

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Liethagoras' Theorem

Stage: 2 and 3

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.

problem icon

Floored

Stage: 3 Challenge Level: Challenge Level:1

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?

problem icon

Trice

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Isosceles

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

problem icon

A Chordingly

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Pythagoras

Stage: 2 and 3

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.

problem icon

Square Pegs

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Some(?) of the Parts

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

The Dangerous Ratio

Stage: 3

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

problem icon

Star Gazing

Stage: 4 Challenge Level: Challenge Level:1

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

problem icon

The Pillar of Chios

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

problem icon

Take a Square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

problem icon

All Is Number

Stage: 2 and 3

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

problem icon

Circumnavigation

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Tennis

Stage: 3 Challenge Level: Challenge Level:1

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?

problem icon

Partly Circles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Tilted Squares

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Tilting Triangles

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Napkin

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Corridors

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

problem icon

Inscribed in a Circle

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Semi-square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Equilateral Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

problem icon

Three Four Five

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

problem icon

Six Discs

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Two Circles

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Fitting In

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

problem icon

Get Cross

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Circle Packing

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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? ...

problem icon

Grid Lockout

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

The Fire-fighter's Car Keys

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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 ?.

problem icon

All Tied Up

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

The Spider and the Fly

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Rhombus in Rectangle

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

problem icon

Cutting a Cube

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Squaring the Circle and Circling the Square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Are You Kidding

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Squ-areas

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

problem icon

Where Is the Dot?

Stage: 4 Challenge Level: Challenge Level:1

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?

problem icon

The Old Goats

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

problem icon

Matter of Scale

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Prove Pythagoras' Theorem using enlargements and scale factors.

problem icon

Medallions

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Ball Packing

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Xtra

Stage: 4 and 5 Challenge Level: Challenge Level:1

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.

problem icon

Semi-detached

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

problem icon

Weighty Problem

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

problem icon

Compare Areas

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

problem icon

Under the Ribbon

Stage: 4 Challenge Level: Challenge Level:1

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


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2017. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.