Calculating exactly with multiples of pi

Three point one four one five nine two six five three five. Pi carries on past that forever. The digits never stop, and they never settle into a repeating pattern. Nobody has ever written pi down completely, and nobody ever will. Which means every decimal version of pi you have ever used - three point one four, or the long string on your calculator screen - is chopped off early. Close to pi. But not pi. In this chapter we have a name for that: decimals leak. Three point one four leaks; pi doesn't. And in this video you'll learn the move mathematicians use to keep circle answers perfectly exact: don't write the digits at all. Write the symbol.

First, the idea itself - and you already know it from algebra. If x stands for a number, you happily write two x, or five x, and carry the x along. You never swap x for a guess at its value. Pi gets exactly the same treatment. Pi is a real, fixed number - a little more than three. But because its decimals leak, we carry the symbol instead of the digits. An answer like nine pi is a finished answer. Not lazy - exact. Quick reminder of the two formulas we'll lean on. The area of a circle: A equals pi r squared. The distance around the outside - the circumference: C equals pi d, pi times the diameter. So take a circle with radius three centimetres. Area equals pi times three squared, which is pi times nine - written as nine pi centimetres squared. Watch what pi did there: it rode through every line as a symbol, exactly like a letter in algebra. And the question itself tells you when to play it this way. Two phrases to spot: 'give your answer in terms of pi', and 'leave your answer as a multiple of pi'. Both mean the same instruction: pi survives to your final answer as a symbol, and no decimal stand-in appears anywhere in your working. If you've seen the video on calculating exactly with fractions, this should feel familiar. Same skill, different costume: spot the instruction, keep the answer exact.

Let's check the idea has landed. A circle has a radius of five centimetres, and the question says 'give your answer in terms of pi'. Three answers on screen. A: seventy eight point five centimetres squared. B: twenty five pi centimetres squared. C: seventy eight point five four centimetres squared. Which one follows the instruction?

Have a think. I'll wait. The answer is B. Twenty five pi is exact - the symbol never leaked. A and C both came from pressing the decimal pi button and rounding, so however many decimal places they keep, they are approximations.

Now, a full exam-style question, start to finish. A circular clock face has a radius of seven centimetres. Work out the exact area of the clock face, giving your answer in terms of pi. Instruction first: 'in terms of pi'. That's the flag. From here on, the decimal pi button is off limits. Write the formula: A equals pi r squared. Substitute the radius: A equals pi times seven squared. Seven squared is forty nine, so A equals forty nine pi. Look down the working: pi sits on every single line. Carried, never converted. Final answer: forty nine pi centimetres squared. And before moving on, look hard at that pi symbol in the final line. Here's why it earns the stare. Forty nine pi is roughly one hundred and fifty four. So if the pi slips off the page and you write plain forty nine, you haven't tidied the answer - you've replaced it with a completely different number, about a third of the size. The pi is not decoration. It's part of the value.

Next, one small step harder - and this one is yours to try. A circular plate has a diameter of ten centimetres. Work out the exact circumference of the plate, leaving your answer as a multiple of pi. You have everything you need. Circumference is pi times the diameter, and 'as a multiple of pi' tells you the symbol survives to the end. Take your time. I'll wait right here. The answer is ten pi centimetres. C equals pi d - pi times ten - ten pi. The sneaky part was the diameter. Plenty of people convert to a radius first - five centimetres - then reach for the calculator just to check, and the screen says thirty one point four one five nine. Copy that down and the leak is in. If you prefer working from the radius, stay symbolic: C equals two pi r, two times pi times five - ten pi again. Symbol in, symbol out.

Time for the examiner's-eye view of this topic. 'In terms of pi' and 'as a multiple of pi' are printed instructions, and your answer is read against them. A rounded decimal - however many places you keep - doesn't follow the instruction, because the exactness leaked away the moment pi became digits. And the news from the marking room is encouraging. One examiner report reads: On a positive note, students seem far more comfortable leaving their answers in terms of pi than on previous series; plus some realised that pi was a common factor and could be cancelled out. Students are getting better at this, and you can join them, because the whole trick is one habit: when the question asks for pi, never press the decimal pi button. If pi never becomes digits, it cannot leak. The second half of that quote is worth a moment too. Carried as a symbol, pi joins in with cancelling, just like a letter does. If a calculation ever hands you forty nine pi over seven pi, the pi cancels top and bottom, leaving forty nine over seven - which is seven. That only works while both pis are still symbols.

One final circle before the recap. Find the exact area of a circle with radius four centimetres, giving your answer as a multiple of pi. And run the honesty test as you work: the decimal version of pi should appear nowhere on your page. Take your time. I'll wait right here. The answer is sixteen pi centimetres squared. A equals pi r squared - pi times four squared - pi times sixteen - sixteen pi. If anything like fifty point two appeared in your working, the leak crept in. If pi sat on every line, you've got this skill.

Let's gather the whole skill into three short lines. One: spot the flag - 'in terms of pi' or 'as a multiple of pi' means the symbol survives to the final line. Two: carry pi like a letter, the way you carry x, and never press the decimal pi button. Three: write the pi in your final answer. Forty nine pi - not forty nine. Or say it the way this chapter says it: decimals leak. Three point one four leaks; pi doesn't.

Next in the chapter: Simplifying surds. A heads-up before you click: that one is a Higher tier topic. If you're sitting Foundation, it goes beyond your paper, so this video is a natural stopping point in the chapter.

For more, visit scholafly.com, or watch the next video.