Other tags that relate to Where to Land
Mathematical reasoning & proof. Visualising. Mathematical modelling. 2D shapes and their properties. Triangles. Pythagoras' theorem. Regular polygons and circles. Sine, cosine, tangent. Creating and manipulating expressions and formulae. Similarity and congruence.There are 65 results
Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties
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.
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?
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.
Towering Trapeziums
Age 14 to 16 Challenge Level:
Can you find the areas of the trapezia in this sequence?
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
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?
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.
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?
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?
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.
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. . . .
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? ...
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.
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?
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.
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.
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?
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?
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.
LOGO Challenge - Circles as Animals
Age 11 to 16 Challenge Level:
See if you can anticipate successive 'generations' of the two animals shown here.
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.
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.
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.
Opposite Vertices
Age 11 to 14 Challenge Level:
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
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.
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. . . .
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!
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. . . .
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.
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.
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?
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.
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?
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?
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. . . .
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?
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.
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. . . .
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?
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?
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.