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There are **49** NRICH Mathematical resources connected to **Circle properties and circle theorems**, you may find related items under Angles, polygons, and geometrical proof.

Problem
Primary curriculum
Secondary curriculum
### Cyclic Quadrilaterals

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circles in Quadrilaterals

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Partly Circles

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Right Angles

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Subtended Angles

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles in Circles

Can you find triangles on a 9-point circle? Can you work out their angles?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Salinon

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Belt

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lens Angle

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sitting Pretty

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Compare Areas

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Baby Circle

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sports Equipment

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Perfect Eclipse

Use trigonometry to determine whether solar eclipses on earth can be perfect.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bicentric Quadrilaterals

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Age 14 to 16

Challenge Level

General
Primary curriculum
Secondary curriculum
### Circle Theorems

This set of resources for teachers offers interactive environments which support work on properties of angles in circles at Key Stage 4.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Interacting with the Geometry of the Circle

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

Age 5 to 16

Interactive
Primary curriculum
Secondary curriculum
### Virtual Geoboard

A virtual geoboard that allows you to create shapes by stretching rubber bands between pegs on the board. Allows a variable number of pegs and variable grid geometry and includes a point labeller.

Problem
Primary curriculum
Secondary curriculum
### Circle Scaling

Describe how to construct three circles which have areas in the ratio 1:2:3.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circle Box

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flower

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lunar Angles

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circles in Circles

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Arrh!

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circumnavigation

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Incircles Explained

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### The Cyclic Quadrilateral

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Crescents and Triangles

Can you find a relationship between the area of the crescents and the area of the triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Similarly So

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tricircle

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 two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Balls

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cyclic Quad Jigsaw

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Encircling

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Four Five

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Not So Little X

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 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Coins on a Plate

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Escriptions

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Strange Rectangle

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Long Short

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circumspection

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Medallions

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Just Touching

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Kissing

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Some(?) of the Parts

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 16

Challenge Level