Year 11 Foundation · Set notation and Venn diagrams
Sort values using overlapping circles
Learn how to find the area and circumference of a circle, work with slices and arcs, and sort values and find probabilities using overlapping circles.
Definition
A circle's radius runs from the centre to the edge, and its diameter runs right across through the centre, so d = 2r. The distance around a circle is its circumference, worked out as π × d, and the space inside is its area, worked out as π × r². A slice of a circle (a sector) is a fraction of the whole one, and its angle tells you exactly what fraction. Circles can also sort information: two overlapping circles hold two groups, and the overlap holds whatever belongs to both.
Why Do I Have to Learn This?
Circles are behind wheels, clocks, coins and orbits. Knowing their parts helps you measure and make round things.
Key Rules:
- 1. In overlapping circles, fill the overlap first with what belongs to both groups, then a probability is the number in the region you want divided by the total.
Remember:
- •Circumference is π × d, area is π × r², and a slice with angle A° takes A/360 of each.
Don’t Forget...
- •Using π × r² with the diameter, or π × d with the radius. With radius 5 cm the area is π × 25, not π × 100: squaring the diameter gives four times the true area, because d = 2r and (2r)² = 4r².
- •Giving only the arc length as the perimeter of a slice. The perimeter is the whole boundary of the slice, so the two straight radii must be added to the arc.
- •Writing the full group totals inside each circle of an overlapping-circles diagram. The 5 who play both already sit in the overlap, so writing 14 in the football-only region counts them twice and the regions no longer add up to the total.
Circle
A flat shape where every point on the edge is the same distance from the centre. Circumference = π × diameter and area = π × radius².
A coin, a clock face and a wheel are all circles.
Worked Example:
In a class of 30 students, 14 play football, 12 play tennis and 5 play both. A student is picked at random. Use overlapping circles to work out the probability that the student plays neither sport.
Put 5 in the overlap first, because those students belong in both circles.
Football only: 14 - 5 = 9. Tennis only: 12 - 5 = 7.
Placed so far: 9 + 5 + 7 = 21, so 30 - 21 = 9 students sit outside both circles.
Probability of neither: 930, which simplifies to 310.
Answer: 310
Why It Works:
π is the same for every circle, so circumference = π × d and area = π × r² hold whatever the size. A slice's angle measures how much of the full 360° turn it uses, and the same share of the turn sweeps out the same share of the area and of the circumference.
Another Example:
A slice of a circle has radius 6 cm and an angle of 60° at the centre. Work out its area and its perimeter, giving each answer to 1 decimal place.
The angle fraction is 60360 = 16, so the slice is 16 of the whole circle.
Area: 16 × π × 6² = 16 × 36π = 6π = 18.849..., which rounds to 18.8 cm².
Arc length: the diameter is 12 cm, so the arc is 16 × π × 12 = 2π = 6.283... cm.
Perimeter: the arc plus the two straight radii, 6.283... + 6 + 6 = 18.283..., which rounds to 18.3 cm.
Answer: Area 18.8 cm² and perimeter 18.3 cm (to 1 d.p.)
Animated Example:
Free Sort values using overlapping circles Worksheet
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Before this: Probability (Year 10 Foundation)
Let’s Practise the Concept
Step 1 of 3
Sort them into groups
Drag each one into the group it belongs in.
Round
Straight sides