# Resources tagged with: Regular polygons and circles

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Broad Topics > Angles, Polygons, and Geometrical Proof > Regular polygons and circles ### Tricircle

##### Age 14 to 16Challenge 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. . . . ### Crescents and Triangles

##### Age 14 to 16Challenge Level

Can you find a relationship between the area of the crescents and the area of the triangle? ##### Age 14 to 16Challenge 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? ### 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. ### Floored

##### Age 14 to 16Challenge 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? ### Lighting up Time

##### Age 7 to 14Challenge Level

A very mathematical light - what can you see? ### Salinon

##### Age 14 to 16Challenge 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? ### Dodecawhat

##### Age 14 to 16Challenge 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. ##### Age 14 to 16Challenge 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. . . . ### Not So Little X

##### Age 11 to 14Challenge 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? ##### Age 14 to 16Challenge Level

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

##### Age 11 to 14Challenge Level

Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring? ##### Age 14 to 16Challenge Level

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex. ### Some(?) of the Parts

##### Age 14 to 16Challenge 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 ### Circumspection

##### Age 14 to 16Challenge 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. ### Fitting In

##### Age 14 to 16Challenge 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. . . . ### LOGO Challenge 1 - Star Square

##### Age 7 to 16Challenge Level

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

##### Age 11 to 14Challenge 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. . . . ### Partly Circles

##### Age 14 to 16Challenge Level

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

##### Age 11 to 14Challenge Level

Which is a better fit, a square peg in a round hole or a round peg in a square hole? ### Efficient Packing

##### Age 14 to 16Challenge Level

How efficiently can you pack together disks? ### LOGO Challenge - Circles as Animals

##### Age 11 to 16Challenge Level

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

##### Age 14 to 16Challenge 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? ... ### Semi-detached

##### Age 14 to 16Challenge 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. ### Get Cross

##### Age 14 to 16Challenge Level

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing? ### 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. ### Hex

##### Age 11 to 14Challenge 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. ### Gibraltar Geometry

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 7 to 16Challenge Level

An environment that enables you to investigate tessellations of regular polygons ### Holly

##### Age 14 to 16Challenge 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. ### Coins on a Plate

##### Age 11 to 14Challenge Level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle. ### Three Four Five

##### Age 14 to 16Challenge 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. ### A Chordingly

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge 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. ### Using Geogebra

##### Age 11 to 18 ### 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. ### Encircling

##### Age 14 to 16Challenge Level

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

##### Age 11 to 14Challenge Level

What fractions of the largest circle are the two shaded regions? ### Pent

##### Age 14 to 18Challenge 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. ### Cube Paths

##### Age 11 to 14Challenge Level

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B? ### F'arc'tion

##### Age 14 to 16 ShortChallenge 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. . . . ### LOGO Challenge 12 - Concentric Circles

##### Age 11 to 16Challenge 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. ### Star Gazing

##### Age 14 to 16Challenge Level

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star. ### The Medieval Octagon

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge 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? ### First Forward Into Logo 2: Polygons

##### Age 7 to 16Challenge Level

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons. ### LOGO Challenge 9 - Overlapping Polygons

##### Age 7 to 16Challenge 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 10 - Circles

##### Age 11 to 16Challenge 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. ### Rolling Around

##### Age 11 to 14Challenge 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?