There are 61 NRICH Mathematical resources connected to 2D shapes and their properties, you may find related items under Angles, Polygons, and Geometrical Proof.Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties
Where should runners start the 200m race so that they have all run the same distance by the finish?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
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?
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.
By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
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. . . .
Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
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.
A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
Can you find the areas of the trapezia in this sequence?
Can Jo make a gym bag for her trainers from the piece of fabric she has?
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
Investigate constructible images which contain rational areas.
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
How efficiently can you pack together disks?
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?
A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?
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.
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. . . .
Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."
Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?
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?
What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?
This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?
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.
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.
See if you can anticipate successive 'generations' of the two animals shown here.
In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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?
Can you find a relationship between the area of the crescents and the area of the triangle?
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.
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. . . .
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.
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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.
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.
How much of the field can the animals graze?
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?
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? ...
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. . . .
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.
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. . . .
For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
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.