Other tags that relate to Trig Reps
Regular polygons and circles. Complex numbers. Sine, cosine, tangent. Pythagoras' theorem. Investigations. Maths Supporting SET. Trigonometric functions and graphs. Maclaurin series. Generalising. Trigonometric identities.There are 68 results
Broad Topics > Angles, Polygons, and Geometrical Proof > Regular polygons and circles
So Big
Age 16 to 18 Challenge Level:
One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2
Circles in Circles
Age 16 to 18 Challenge Level:
This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.
Spokes
Age 16 to 18 Challenge Level:
Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
Just Touching
Age 16 to 18 Challenge Level:
Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?
A Rational Search
Age 16 to 18 Challenge Level:
Investigate constructible images which contain rational areas.
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?
Lunar Angles
Age 16 to 18 Challenge Level:
What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?
Gold Yet Again
Age 16 to 18 Challenge Level:
Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."
Escriptions
Age 16 to 18 Challenge Level:
For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
Using Geogebra
Age 11 to 18
Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.
Ford Circles
Age 16 to 18 Challenge Level:
Can you find the link between these beautiful circle patterns and Farey Sequences?
Ball Bearings
Age 16 to 18 Challenge Level:
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
Sangaku
Age 16 to 18 Challenge Level:
The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.
Area I'n It
Age 16 to 18 Challenge Level:
Triangle ABC has altitudes h1, h2 and h3. The radius of the inscribed circle is r, while the radii of the escribed circles are r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1 +. . . .
Retracircles
Age 16 to 18 Challenge Level:
Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.
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?
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 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.
LOGO Challenge - Circles as Animals
Age 11 to 16 Challenge Level:
See if you can anticipate successive 'generations' of the two animals shown here.
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?
Incircles Explained
Age 16 to 18
This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.
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.
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.
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.
Tessellation Interactivity
Age 7 to 16 Challenge Level:
An environment that enables you to investigate tessellations of regular polygons
Spirostars
Age 16 to 18 Challenge Level:
A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?
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.
2D-3D
Age 16 to 18 Challenge Level:
Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of. . . .
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?
Logosquares
Age 16 to 18 Challenge Level:
Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.
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.
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.
Baby Circle
Age 16 to 18 Challenge Level:
A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
Orthogonal Circle
Age 16 to 18 Challenge Level:
Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
Kissing
Age 16 to 18 Challenge Level:
Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
The Dodecahedron
Age 16 to 18 Challenge Level:
What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?
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?
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?
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.
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?
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.
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. . . .
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
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.
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? ...
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
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 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.
NRICH is part of the family of activities in the Millennium Mathematics Project.