Resources tagged with: Regular polygons and circles

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

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 6 - Triangles and Stars

Age 11 to 16Challenge 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.

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 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.

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?

LOGO Challenge 11 - More on Circles

Age 11 to 16Challenge 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 16Challenge Level

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

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.

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.

LOGO Challenge - Circles as Animals

Age 11 to 16Challenge Level

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

Age 7 to 14Challenge Level

What shape and size of drinks mat is best for flipping and catching?

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. . . .

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.

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. . . .

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. . . .

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?

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?

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?

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?

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.

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.

Two Regular Polygons

Age 14 to 16Challenge 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?

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. . . .

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

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.

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?

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.

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?

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.

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.

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.

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? ...

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?

Partly Circles

Age 14 to 16Challenge Level

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

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?

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?

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.

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?

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?

Bull's Eye

Age 11 to 14Challenge Level

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

Tessellation Interactivity

Age 7 to 16Challenge Level

An environment that enables you to investigate tessellations of regular polygons

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.

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?

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.

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.

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.