# Resources tagged with: Regular polygons and circles

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There are 61 NRICH Mathematical resources connected to Regular polygons and circles, you may find related items under Angles, Polygons, and Geometrical Proof.

Broad Topics > Angles, Polygons, and Geometrical Proof > Regular polygons and circles ### 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? ### Partly Circles

##### Age 14 to 16 Challenge Level:

What is the same and what is different about these circle questions? What connections can you make? ### 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? ### 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-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. ### Approximating Pi

##### Age 14 to 18 Challenge Level:

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? ### 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. ##### 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. . . . ### Shogi Shapes

##### Age 11 to 14 Challenge Level:

Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring? ### Gibraltar Geometry

##### Age 11 to 14 Challenge Level:

Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them? ### Using Geogebra

##### Age 11 to 18 ##### Age 7 to 14 Challenge Level:

What shape and size of drinks mat is best for flipping and catching? ##### 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. ### Efficient Packing

##### Age 14 to 16 Challenge Level:

How efficiently can you pack together disks? ### Tessellation Interactivity

##### Age 7 to 16 Challenge Level:

An environment that enables you to investigate tessellations of regular polygons ### Lighting up Time

##### Age 7 to 14 Challenge Level:

A very mathematical light - what can you see? ### Two Regular Polygons

##### Age 14 to 16 Challenge Level:

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? ### 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? ### LOGO Challenge 12 - Concentric Circles

##### Age 11 to 16 Challenge Level:

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. ### LOGO Challenge 11 - More on Circles

##### Age 11 to 16 Challenge Level:

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. . . . ### 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? ### First Forward Into Logo 2: Polygons

##### Age 7 to 16 Challenge Level:

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons. ##### Age 14 to 16 Challenge Level:

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? ### 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. ### 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. ##### 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. ### Dodecawhat

##### Age 14 to 16 Challenge Level:

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. ### LOGO Challenge - Following On

##### Age 11 to 18 Challenge Level:

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? ### LOGO Challenge 9 - Overlapping Polygons

##### Age 7 to 16 Challenge Level:

This LOGO challenge starts by looking at 10-sided polygons then generalises the findings to any polygon, putting particular emphasis on external angles ### LOGO Challenge 6 - Triangles and Stars

##### Age 11 to 16 Challenge Level:

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. ### LOGO Challenge - Circles as Animals

##### Age 11 to 16 Challenge Level:

See if you can anticipate successive 'generations' of the two animals shown here. ### LOGO Challenge 10 - Circles

##### Age 11 to 16 Challenge Level:

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further. ### LOGO Challenge 1 - Star Square

##### Age 7 to 16 Challenge Level:

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed. ### 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. ### 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. ### 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. . . . ### 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? ### 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? ### 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. ### Bull's Eye

##### Age 11 to 14 Challenge Level:

What fractions of the largest circle are the two shaded regions? ### 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. . . . ### F'arc'tion

##### Age 14 to 16 Short Challenge Level:

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. . . . ### 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. ### Encircling

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle. ### A Chordingly

##### Age 11 to 14 Challenge Level:

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. ### 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?