# Resources tagged with: Regular polygons and circles

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

##### Age 14 to 16Challenge Level

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon? ### 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? ### Squaring the Circle and Circling the Square

##### Age 14 to 16Challenge Level

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction. ### 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. ### Like a Circle in a Spiral

##### Age 7 to 16Challenge 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? ##### 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? ### 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? ### 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. ### LOGO Challenge - Circles as Animals

##### Age 11 to 16Challenge Level

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

##### Age 7 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

How efficiently can you pack together disks? ### 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. ### 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. ### 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. ### 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? ### 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? ### 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? ### 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? ### Using Geogebra

##### Age 11 to 18 ### 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. ### 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? ### 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? ### LOGO Challenge - Following On

##### Age 11 to 18Challenge 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? ### 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

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. . . . ##### 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. ### 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. . . . ### 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. ##### 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 ### 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? ### 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? ... ### 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? ### 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. ### Lighting up Time

##### Age 7 to 14Challenge Level

A very mathematical light - what can you 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. ### 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. ### 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? ### 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? ### Bull's Eye

##### Age 11 to 14Challenge Level

What fractions of the largest circle are the two shaded regions? ### 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. . . . ### From All Corners

##### Age 14 to 16Challenge Level

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. ### Partly Circles

##### Age 14 to 16Challenge Level

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

##### Age 14 to 18Challenge 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? ### 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. ### 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. . . . ### 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. ### First Forward Into Logo 4: Circles

##### Age 7 to 16Challenge Level

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