There are **61** NRICH Mathematical resources connected to **2D shapes and their properties**, you may find related items under Angles, Polygons, and Geometrical Proof.

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Where should runners start the 200m race so that they have all run the same distance by the finish?

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

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

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Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

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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?

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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.

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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?

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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. . . .

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Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

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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.

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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?

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

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

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

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Investigate constructible images which contain rational areas.

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

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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?

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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?

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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.

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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. . . .

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Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

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

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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?

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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?

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This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

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The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

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

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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.

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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.

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

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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.

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If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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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?

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

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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.

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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. . . .

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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.

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

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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.

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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.

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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?

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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? ...

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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. . . .

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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.

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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. . . .

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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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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.