Other tags that relate to Generating Triples
Circle properties and circle theorems. Square numbers. Regular polygons and circles. Mathematical reasoning & proof. Making and proving conjectures. Generalising. Polynomial functions and their roots. Creating and manipulating expressions and formulae. Pythagoras' theorem. Properties of numbers.There are 61 results
Broad Topics > Angles, Polygons, and Geometrical Proof > Regular polygons and circles
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?
Coins on a Plate
Age 11 to 14 Challenge 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.
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?
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.
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.
Polycircles
Age 14 to 16 Challenge 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?
Fitting In
Age 14 to 16 Challenge 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. . . .
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?
Rolling Coins
Age 14 to 16 Challenge Level:
A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .
Circumnavigation
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.
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?
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.
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.
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?
Square Pegs
Age 11 to 14 Challenge Level:
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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?
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?
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?
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?
Quadarc
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. . . .
Circle Packing
Age 14 to 16 Challenge 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 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.
Bicentric Quadrilaterals
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.
Some(?) of the Parts
Age 14 to 16 Challenge 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 16 Challenge 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?
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. . . .
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.
Star Gazing
Age 14 to 16 Challenge Level:
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
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?
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.
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.
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.
Circumspection
Age 14 to 16 Challenge 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.
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.
Floored
Age 11 to 14 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?
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.
Cube Paths
Age 11 to 14 Challenge 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?
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?
Tessellation Interactivity
Age 7 to 16 Challenge Level:
An environment that enables you to investigate tessellations of regular polygons
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. . . .
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. . . .
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.
From All Corners
Age 14 to 16 Challenge 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.
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 - Circles as Animals
Age 11 to 16 Challenge Level:
See if you can anticipate successive 'generations' of the two animals shown here.
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 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
NRICH is part of the family of activities in the Millennium Mathematics Project.