Other tags that relate to Where to Land
Mathematical reasoning & proof. Sine, cosine, tangent. Similarity and congruence. Ratio and proportion. 2D shapes and their properties. Creating and manipulating expressions and formulae. Mathematical modelling. Pythagoras' theorem. Visualising. Regular polygons and circles.There are 65 results
Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties
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.
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?
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? ...
Towering Trapeziums
Age 14 to 16
Challenge Level
Can you find the areas of the trapezia in this sequence?
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. . . .
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.
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?
Pent
Age 14 to 18
Challenge Level
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
From All Corners
Age 14 to 16
Challenge 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.
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.
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.
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?
Salinon
Age 14 to 16
Challenge 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?
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.
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.
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.
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
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.
Trapezium Four
Age 14 to 16
Challenge 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?
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?
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?
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.
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?
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.
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.
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.
Rolling Around
Age 11 to 14
Challenge Level
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Playground Snapshot
Age 7 to 14
Challenge 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?
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.
LOGO Challenge - Circles as Animals
Age 11 to 16
Challenge Level
See if you can anticipate successive 'generations' of the two animals shown here.
From One Shape to Another
Age 7 to 14
Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.
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.
Blue and White
Age 11 to 14
Challenge Level
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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. . . .
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. . . .
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?
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!
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.
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. . . .
Opposite Vertices
Age 11 to 14
Challenge Level
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
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?
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?
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.
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?
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?
NRICH is part of the family of activities in the Millennium Mathematics Project.