What are some interesting properties of circles?

Key questions

  1. 1

    How are circles and angle measurement related?

  2. 2

    How can we find the arc length and area of sectors of circles?

  3. 3

    How can we use the Cartesian equation of a circle?

  4. 4

    How are circle theorems useful?

  5. 5

    In what ways can circles be generalised?

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Resource type Title
Fluency exercise Sector spirals


Resource type Title
Building blocks Cones
Package of problems Elliptical crossings
Package of problems Pairs of circles
Package of problems Teddy bear
Many ways problem Finding circles
Problem requiring decisions Olympic rings
Food for thought Belt
Food for thought Cutting spheres
Bigger picture Conic sections in real life
Bigger picture Where are you?
Resource in action Teddy bear - teacher support

Review questions

Title Ref
Can we find the area between three touching circles? R8225
Can we find the area bounded by these two graphs? R9386
Can we find the areas enclosed between a circle and its tangent? R8946
Can we find the circles through the points where two ellipses meet? R8216
Can we find the equation of the inscribed circle of the triangle $ABC$? R5777
Can we find the equations of two circles that touch another? R5624
Can we find the ratio of the areas of these two sectors? R7816
Can we find the tangents to this circle from the origin? R5903
Can we find the two circles that satisfy these three conditions? R8419
Can we show that (circle area)/(triangle area) $\geq \pi$? R7656
Can we show that these four points lie on a circle? R5717
Can we show that these two circles touch? R6100
Can we show the locus of the midpoint of PQ is an ellipse? R8051
Can we show these two circles lie entirely outside each other? R6390
Can we show this quartic equation has exactly two real roots? R7549
Given three clues, can we find the equation of this circle? R9368
Given two lines crossing a circle, can we show these lengths are equal? R9551
If $AQ:QP=\lambda:1$, can we show $Q$ lies on a circle? R9045
If $x^2+y^2\leq1$, when does $x+y$ have a maximum? R6645
If we know the difference in their areas, how big are these sectors? R9423
What area do these intersecting circles share? R7937
What can we deduce if the loci traced out by two points touch? R8509
What can we say if two circles cut at right angles? R7925
What's the area inside both the circle $C$ and the triangle $T$? R7495
What's the area of this square if it sits in this sector? R6164
What's the locus of $(p, q)$ if the length of this tangent equals $p$? R9768
What's the shared area for these two circles? R6838
When does a circle touch a parabola twice? R7979
When does the origin lie inside this circle? R7599
When does this area within a circle have a maximum value? R9028
Where do this pair of tangents to a circle meet? R7171
Which point on the first circle is closest to the second? R9452
Which point on this circle is closest to the origin? R8275