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

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Pythagoras Mod 5

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

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Grid Lockout

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

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

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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 ?

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Pythagoras for a Tetrahedron

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . .

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Picturing Pythagorean Triples

Stage: 4 and 5

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.

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Pythagorean Triples I

Stage: 3 and 4

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!

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Round and Round

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

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

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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. . . .

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Pythagorean Triples II

Stage: 3 and 4

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

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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. . . .

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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?

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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?

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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? ...

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Retracircles

Stage: 5 Challenge Level: Challenge Level:1

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

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Rectangular Pyramids

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

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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Holly

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

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.

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Some(?) of the Parts

Stage: 4 Challenge Level: Challenge Level:1

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

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Medallions

Stage: 4 Challenge Level: Challenge Level:1

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized. . . .

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Logosquares

Stage: 5 Challenge Level: Challenge Level:1

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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Square World

Stage: 5 Challenge Level: Challenge Level:1

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

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Reach for Polydron

Stage: 5 Challenge Level: Challenge Level:1

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

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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.

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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.

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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.

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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?

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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.

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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.

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Strange Rectangle

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

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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.

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Chord

Stage: 5 Challenge Level: Challenge Level:1

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

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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.

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Kite in a Square

Stage: 4 Challenge Level: Challenge Level:1

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

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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 ?.

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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?

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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?

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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?

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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?

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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.

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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?

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Circle Scaling

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

You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.

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Circle Box

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

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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Ladder and Cube

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

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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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.

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Little and Large

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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Pythagoras Proofs

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

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

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Square Pair Circles

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

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Zig Zag

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

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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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 .

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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?

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30-60-90 Polypuzzle

Stage: 5 Challenge Level: Challenge Level:1

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.