GCSE · Maths · AQA · Spec 8300 · Foundation
Circle definitions and properties
Most people call any line across a circle a diameter, and most of the time they are wrong. Move one line and watch when it earns the name.
Circles · One line, three names
Every radius is the same length. A chord through the centre is a diameter: the longest chord, and exactly two radii long. A line that touches the circle at exactly one point is a tangent.
Drag B round the circle: radius OB always reads 5.0 cm, and chord AB is longest, 10.0 cm, when it passes through the centre O. Then drag P. While line PQ cuts the circle, the part inside is a chord, and a diameter when it passes through O. At one position PQ just touches the circle at a single point: it is a tangent.
Circles · What kind of thing is it?
Point, line, curve or region?
What kind of object does each word name? Pick a word, then pick its group.
Still to sort
A point (0)
A single position with no length.
A straight line (0)
Something you could draw with a ruler.
Where the line is: A curve along the edge is not a straight line, even though it joins two points.
A curve on the edge (0)
Some or all of the circle's boundary.
Where the line is: A curve has length but no area. The moment you include the space inside, it becomes a region.
A region (0)
A piece of the space inside the circle.
Where the line is: A region is always bounded by lines or curves, but it is the area inside them, not the boundary itself.
Nine circle words, four kinds of object. Decide what kind each one is before you read why.
Using the definitions
Problem
O is the centre of a circle with diameter 16 cm. P and Q are points on the circumference, and angle POQ = 40°. (a) Work out the length of OP. (b) Work out the size of angle OPQ.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Every part of a circle gets its name from where it sits.
What you need to know
- The centre is the point that is the same distance from every point on the circle.
- A radius joins the centre to the circumference. Every radius of a circle is the same length.
- A chord joins two points on the circumference. A diameter is a chord that passes through the centre.
- A diameter is two radii long (d = 2r), which makes it the longest chord in the circle.
- The circumference is the whole edge of the circle, and an arc is part of it.
- A tangent is a straight line that touches the circle at exactly one point.
- A sector is the region between two radii and an arc. A segment is the region between a chord and an arc.
The big picture
Each part of a circle is named by where it sits. A radius joins the centre to the edge, and every radius is the same length. A chord joins two points on the edge; the chord through the centre is the diameter, twice a radius and the longest chord. The circumference is the whole edge and an arc is part of it. A tangent touches the circle at exactly one point. A sector is cut off by two radii and an arc, a segment by a chord and an arc.
Key points
Worked example
Problem
A circle has radius 4.5 cm. PQ is a chord of the circle and is 7 cm long. (a) Work out the length of the diameter. (b) Does the chord PQ pass through the centre of the circle? Give a reason.
⚠ Watch out
Calling any line across a circle a diameter. A line from one side of the circle to the other is a chord. It is only a diameter if it passes through the centre, and every other chord is shorter than the diameter.
Memory hook
Radius reaches, chord crosses, diameter goes through the middle, tangent just touches. A sector is a pizza slice with its point at the centre. A segment is the piece one straight cut takes off.
Check yourself
Sketch a circle and label a radius, a chord that is not a diameter, a tangent, an arc, a sector and a segment. If the radius is 6.5 cm, how long is the longest chord?
Flashcards
(12)What is the centre of a circle?
What is a radius?
What is a chord?
What is a diameter?
What is the circumference?
What is a tangent?
What is an arc?
What is a sector?
What is a segment?
How is the length of a diameter related to the length of a radius?
O is the centre of a circle, and A and B are on the circle. Why must triangle OAB be isosceles?
What do 'minor' and 'major' mean for sectors and segments?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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