# Resources tagged with: 2D shapes and their properties

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### There are 65 results

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

### Opposite Vertices

##### Age 11 to 14Challenge Level

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

### Lawnmower

##### Age 14 to 16Challenge 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. . . .

### Square Areas

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

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

### Towering Trapeziums

##### Age 14 to 16Challenge Level

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

### Circumspection

##### Age 14 to 16Challenge 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.

##### Age 14 to 16Challenge 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. . . .

### Dividing the Field

##### Age 14 to 16Challenge 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. . . .

### LOGO Challenge - Circles as Animals

##### Age 11 to 16Challenge Level

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

### Not So Little X

##### Age 11 to 14Challenge 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?

### Semi-detached

##### Age 14 to 16Challenge 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.

### 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.

### Fitting In

##### Age 14 to 16Challenge 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. . . .

### 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.

##### Age 14 to 16Challenge 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 14Challenge 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.

### Salinon

##### Age 14 to 16Challenge Level

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

### Efficient Packing

##### Age 14 to 16Challenge Level

How efficiently can you pack together disks?

### LOGO Challenge 12 - Concentric Circles

##### Age 11 to 16Challenge 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 11 - More on Circles

##### Age 11 to 16Challenge 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 6 - Triangles and Stars

##### Age 11 to 16Challenge 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.

### Trapezium Four

##### Age 14 to 16Challenge Level

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

### What Shape?

##### Age 7 to 14Challenge 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.

### Curvy Areas

##### Age 14 to 16Challenge Level

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

### What Shape for Two

##### Age 7 to 14Challenge Level

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

### 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.

### Circle Packing

##### Age 14 to 16Challenge 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? ...

### Square Pegs

##### Age 11 to 14Challenge Level

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

### LOGO Challenge 10 - Circles

##### Age 11 to 16Challenge 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.

### Crescents and Triangles

##### Age 14 to 16Challenge Level

Can you find a relationship between the area of the crescents and the area of the triangle?

### Tricircle

##### Age 14 to 16Challenge 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. . . .

### Rhombus in Rectangle

##### Age 14 to 16Challenge 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.

### 2001 Spatial Oddity

##### Age 11 to 14Challenge 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.

### Holly

##### Age 14 to 16Challenge 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.

### Three Four Five

##### Age 14 to 16Challenge 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.

### Tied Up

##### Age 14 to 16 ShortChallenge Level

How much of the field can the animals graze?

### Floored

##### Age 14 to 16Challenge 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 16Challenge 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

### Squaring the Circle

##### Age 11 to 14Challenge 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. . . .

### Blue and White

##### Age 11 to 14Challenge Level

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

### Lying and Cheating

##### Age 11 to 14Challenge 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!

### Poly Plug Rectangles

##### Age 5 to 14Challenge 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?

### Roaming Rhombus

##### Age 14 to 16Challenge 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. . . .

### From All Corners

##### Age 14 to 16Challenge Level

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### First Forward Into Logo 4: Circles

##### Age 7 to 16Challenge Level

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

### Pentagonal

##### Age 14 to 16Challenge 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 14Challenge 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.

### Playground Snapshot

##### Age 7 to 14Challenge Level

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

### Like a Circle in a Spiral

##### Age 7 to 16Challenge 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?

### The Medieval Octagon

##### Age 14 to 16Challenge Level

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