Resources tagged with: 2D shapes and their properties

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Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties

First Forward Into Logo 4: Circles

Age 7 to 16 Challenge Level:

Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?

LOGO Challenge 8 - Rhombi

Age 7 to 16 Challenge Level:

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

LOGO Challenge 2 - Diamonds Are Forever

Age 7 to 16 Challenge Level:

The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

LOGO Challenge 10 - Circles

Age 11 to 16 Challenge Level:

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.

LOGO Challenge 6 - Triangles and Stars

Age 11 to 16 Challenge Level:

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

LOGO Challenge 11 - More on Circles

Age 11 to 16 Challenge Level:

Thinking of circles as polygons with an infinite number of sides - but how does this help us with our understanding of the circumference of circle as pi x d? This challenge investigates. . . .

LOGO Challenge 12 - Concentric Circles

Age 11 to 16 Challenge Level:

Can you reproduce the design comprising a series of concentric circles? Test your understanding of the realtionship betwwn the circumference and diameter of a circle.

LOGO Challenge - Circles as Animals

Age 11 to 16 Challenge Level:

See if you can anticipate successive 'generations' of the two animals shown here.

What Shape for Two

Age 7 to 14 Challenge Level:

'What Shape?' activity for adult and child. Can you ask good questions so you can work out which shape your partner has chosen?

Arclets Explained

Age 11 to 16

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.

Circles, Circles Everywhere

Age 7 to 14

This article for pupils gives some examples of how circles have featured in people's lives for centuries.

Darts and Kites

Age 14 to 16 Challenge Level:

Explore the geometry of these dart and kite shapes!

Bow Tie

Age 11 to 14 Challenge Level:

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

Tricircle

Age 14 to 16 Challenge Level:

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

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.

Opposite Vertices

Age 11 to 14 Challenge Level:

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Roaming Rhombus

Age 14 to 16 Challenge Level:

We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point. . . .

What's Inside/outside/under the Box?

Age 7 to 14

This article describes investigations that offer opportunities for children to think differently, and pose their own questions, about shapes.

Not So Little X

Age 11 to 14 Challenge Level:

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

Square Areas

Age 11 to 14 Challenge Level:

Can you work out the area of the inner square and give an explanation of how you did it?

What Shape?

Age 7 to 14 Challenge Level:

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

Quadarc

Age 14 to 16 Challenge Level:

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .

Lawnmower

Age 14 to 16 Challenge Level:

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of. . . .

Circumspection

Age 14 to 16 Challenge Level:

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Squaring the Circle

Age 11 to 14 Challenge Level:

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

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.

Pi, a Very Special Number

Age 7 to 14

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

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. . . .

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? ...

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?

2001 Spatial Oddity

Age 11 to 14 Challenge Level:

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

Bicentric Quadrilaterals

Age 14 to 16 Challenge Level:

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Lying and Cheating

Age 11 to 14 Challenge Level:

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

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.

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?

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

Gym Bag

Age 11 to 16 Challenge Level:

Can Jo make a gym bag for her trainers from the piece of fabric she has?

Towering Trapeziums

Age 14 to 16 Challenge Level:

Can you find the areas of the trapezia in this sequence?

Witch's Hat

Age 11 to 16 Challenge Level:

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

Curvy Areas

Age 14 to 16 Challenge Level:

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Dividing the Field

Age 14 to 16 Challenge Level:

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

Efficient Packing

Age 14 to 16 Challenge Level:

How efficiently can you pack together disks?

Poly Plug Rectangles

Age 5 to 14 Challenge Level:

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

Tied Up

Age 14 to 16 Short Challenge Level:

How much of the field can the animals graze?

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.

Polycircles

Age 14 to 16 Challenge Level:

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Pentagonal

Age 14 to 16 Challenge Level:

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

Efficient Cutting

Age 11 to 14 Challenge Level:

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

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.

Like a Circle in a Spiral

Age 7 to 16 Challenge Level:

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?