There are **68** NRICH Mathematical resources connected to **Regular polygons and circles**, you may find related items under Angles, Polygons, and Geometrical Proof.

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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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What is the same and what is different about these circle questions? What connections can you make?

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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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One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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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 circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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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?

Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.

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Investigate constructible images which contain rational areas.

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Can you find the link between these beautiful circle patterns and Farey Sequences?

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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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An environment that enables you to investigate tessellations of regular polygons

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Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

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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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This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

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Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?

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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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What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

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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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Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

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Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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This LOGO challenge starts by looking at 10-sided polygons then generalises the findings to any polygon, putting particular emphasis on external angles

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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.

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Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

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.

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

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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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At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

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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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An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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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.