GCSE · Maths · AQA · Spec 8300 · Foundation
Exact calculation with multiples of pi
π's decimals never end, so how can an answer with π in it ever be exact? Here's the trick: don't turn π into a decimal at all.
How π behaves
What kind of answer do you get?
Pick a calculation, then the box its simplified answer belongs in.
Still to sort
A multiple of π (0)
The answer looks like 8π.
Where the line is: Adding, subtracting or multiplying by an ordinary number keeps exactly one π. Multiplying by another π-term does not.
A multiple of π² (0)
The answer looks like 12π².
Where the line is: Count the π's being multiplied together: two π's make π². Adding two π-terms never does.
No π left (0)
The answer is an ordinary number.
Where the line is: π disappears only when a π on top is cancelled by a π underneath. Dividing by an ordinary number leaves it alone.
Stays as two terms (0)
Nothing can be combined.
Where the line is: π-terms combine only with π-terms, and π²-terms only with π²-terms. Anything else is already as simple as it gets.
Decide before you work it out. The kind of answer tells you which rule you're using.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
π is one fixed number. Count in it, combine it, cancel it, and your answer stays exact.
What you need to know
- π is a fixed number, a little over 3 (3.14159…). Its decimals go on for ever without repeating, so any decimal you write for π, such as 3.14, is a rounded value.
- Calculating exactly with multiples of π means keeping π as a symbol. An exact answer looks like 8π or 9π², not 25.13 or 88.83.
- Multiples of π add and subtract like terms in algebra: combine the numbers in front and keep the π. π²-terms combine with π²-terms in the same way.
- To multiply, deal with the numbers and the π's separately. π × π = π², so a π-term times a π-term gives a multiple of π².
- To divide, cancel matching π's: 8π ÷ 2π = 4, while 8π ÷ 2 = 4π. A π-term and an ordinary number, such as 6π + 1, cannot be combined.
- Only use a decimal for π if a question asks for a rounded answer, and then only at the very last step.
The big picture
π is one fixed number, a little over 3, and its decimals never end. To calculate exactly, you keep π as a symbol and count in it: like π-terms add and subtract, π × π = π², and a π-term divided by a π-term leaves no π. Swap π for 3.14 at any point and the answer becomes approximate.
Key points
Worked example
Problem
Work out 5π × 6π ÷ 3π − 4π. Give your answer in terms of π.
⚠ Watch out
Letting π merge with anything. 2π + 3 is not 5π, because a π-term and an ordinary number are not like terms, and π × π is π², not π or 2π.
Memory hook
π stays π: count it, combine it, cancel it. Never swap it for 3.14.
Check yourself
Without looking back: simplify 7π − 2π + 4 as far as possible, and explain why you can't go any further.
Flashcards
(6)What does it mean to give an answer exactly, in terms of π?
Why is 3.14 not the same as π?
How do you add or subtract multiples of π?
What is π × π?
What happens when you divide a π-term by a π-term, such as 15π ÷ 5π?
Can 5π + 1 be simplified?
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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