Resources tagged with: Regular polygons and circles

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

A Rational Search

Age 16 to 18
Challenge Level

Investigate constructible images which contain rational areas.

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.

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.

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.

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?

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?

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.

Gold Yet Again

Age 16 to 18
Challenge Level

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

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?

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?

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.

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.

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.

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?

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?

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.

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?

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.

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.

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. . . .

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

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.

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? ...

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. . . .

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?

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.

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?

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.

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.

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.

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?

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?

LOGO Challenge 10 - Circles

Age 11 to 16
Challenge 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.

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. . . .

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 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

Ford Circles

Age 16 to 18
Challenge Level

Can you find the link between these beautiful circle patterns and Farey Sequences?

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.

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.

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.

Pegboard Quads

Age 14 to 16
Challenge 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?

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. . . .