Recalling Circle, Pythagoras and Trig Formulae

Picture a rectangular park between you and the shops, fifty metres along one edge and a hundred and twenty along the other. A worn path runs straight across the grass, corner to corner, because nobody wants to walk round two sides. Walking the two edges is a hundred and seventy metres. Straight across is a hundred and thirty. That worn path saves forty metres of walking, and you can work that number out without pacing a single step of it. Look again at the hundred and seventy, because it is nothing more than the two edges added together. Add the sides of a right-angled triangle and you have not found the way across at all. You have described the long way round.

This video is about three families of formulae you have to carry in your head: circles, Pythagoras, and the right-angled trigonometry ratios.

OCR hands you no formula sheet for any of them. The circle formulae, Pythagoras and the ratios are pure recall, so if you cannot write the formula down, the question never even starts. That is why this is drilling rather than reading.

Start with the circle, because it has two formulae and they are easy to swap over. Circumference is the distance all the way round the edge, and it is pi times the diameter. If the radius is what you were given instead, it is two times pi times the radius, because the diameter is twice the radius. Area is the space inside, and that one is pi times the radius squared. Only one of these two formulae carries a square, and it is the area. There is a reason behind that, not just a rule to trust. Area is always measured in squared units, square centimetres or square metres. So the formula that hands you an area is the one with the squared radius in it. Squared units, squared formula.

Quick check, and you can mark this one yourself. A circle has radius r. Which expression gives the distance round the edge: two pi r, or pi r squared?

Pick one. I'll wait.

The distance round the edge is two pi r. Pi r squared is the one carrying the square, so that expression is the area. If you went the other way, say it once more: squared units, squared formula.

Now put it to work. A circular plate has a diameter of ten centimetres. Work out the circumference, giving your answer in terms of pi. The diameter is what you were handed, so the circumference is pi times ten, which you write as ten pi centimetres. In terms of pi means the pi symbol stays standing in your answer. Do not reach for the pi button and write thirty one point four. When a question wants a decimal it asks for one, in decimal places or significant figures. Ten pi centimetres is the finished answer.

On to the triangle now, and the theorem that finds the way across that park.

Pythagoras' theorem works on right-angled triangles, and only on right-angled triangles. The longest side is the hypotenuse, and it always sits opposite the right angle, never touching it. The theorem itself is a squared plus b squared equals c squared. C is the hypotenuse, a and b are the two shorter sides, and it makes no difference which of those two you call which. The squares are the whole point, so here is what they actually mean. Build a real square on each shorter side, and a real square on the hypotenuse. The areas of the two smaller squares add up to exactly the area of the big one. The theorem is a statement about areas, and that is precisely why you square before you add.

Your turn, on the triangle from the start. A right-angled triangle has two shorter sides of five centimetres and twelve centimetres. Find the length of the hypotenuse.

Pause it there and work it out. I'll wait.

Every step in full. Five squared is twenty five. Twelve squared is one hundred and forty four. Twenty five plus one hundred and forty four is one hundred and sixty nine. That total is c squared, not c, so take the square root of it. The hypotenuse is thirteen centimetres.

About that last step. If your square numbers are solid up to around fifteen, one hundred and sixty nine is thirteen squared on sight. If they are not, the square root button hands you the same thirteen.

Now the answer to avoid. Add five and twelve without squaring and you get seventeen, which is the two sides walked end to end. Seventeen is the long way round the corner, so it cannot also be the short way across. A hypotenuse is always less than the other two sides added together.

Three words hold the order for you: square, add, root. Square both of the sides you know, add them together, then root the total.

One variation to know. If the hypotenuse is the side you already have, you subtract instead of adding, because c squared is the whole total and a shorter side is only part of it.

Third family now, and this is the one that needs an angle.

Pythagoras has no angle in it anywhere, so when a question hands you one angle and one side, it cannot help. Sine, cosine and tangent are the three ratios linking an angle to the sides. Label the sides from where the angle is standing. The hypotenuse is the longest side, opposite the right angle, and it never moves. The opposite side faces your angle across the triangle, and the adjacent side is the one touching it. Stand at the other angle instead, and opposite and adjacent swap over. Then the three ratios themselves. Sine of the angle is opposite over hypotenuse. Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. The old chant holds all three in the right order. S, O, H: sine, opposite, hypotenuse. C, A, H: cosine, adjacent, hypotenuse. T, O, A: tangent, opposite, adjacent.

One more for you. A right-angled triangle has an angle of thirty degrees and a hypotenuse of eight centimetres. You want the side opposite that thirty degree angle. Which ratio uses opposite and hypotenuse: sine, cosine, or tangent?

Take your pick. I'll wait.

Sine is the one, because S, O, H is sine, opposite, hypotenuse. So sine of thirty degrees equals the opposite side divided by eight.

Finish it off from there. Sine of thirty degrees is exactly one half, and that value is worth knowing by heart. Half of eight is four, so the side opposite the thirty degree angle is four centimetres.

One setting to check before you trust any of that. Your calculator has to be in degrees mode. Left in radians, sine of thirty comes out as minus nought point nine nine, and the whole answer goes with it.

Now to what examiners write in their reports after marking questions like these.

Here is one line from an examiner report, about a question where Pythagoras' theorem had to be used.

Candidates that are more able understood Pythagoras' theorem but less able ones often added or subtracted sides without squaring.

Added or subtracted sides without squaring. That is our seventeen, turning up on real papers, written by students who knew the theorem existed. Squaring is not a formality on the way to the answer. Squaring is the theorem.

And a line from a second report, on a question where the area of a circle had to be found.

Students still confuse the formulae for the area and the circumference of a circle, thus only around half the students gained the 2 marks here for finding the area of a circle.

Confuse is the precise word there. Both formulae were probably sitting in their heads, and the wrong one came out onto the page.

So here is the check that actually matters, and it is not a calculation. With nothing in front of you, write down four things from memory: circumference, area of a circle, Pythagoras' theorem, and the three ratios, every letter correctly labelled.

Take your time. I'll wait right here.

Mark yourself against these. Circumference is pi times the diameter. Area is pi times the radius squared. A squared plus b squared equals c squared. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Whichever one you could not write down is tonight's revision.

That is what recall actually means here: no sheet, no prompt, and no hesitation before your pen starts moving.

Running the whole thing back now. Circles: circumference is pi times the diameter, area is pi times the radius squared, and squared units mean the squared formula. Pythagoras: right-angled triangles only, hypotenuse opposite the right angle, and square, add, root, in that order. Adding without squaring hands you the long way round instead. Trigonometry: label opposite, adjacent and hypotenuse from your angle, then S, O, H, C, A, H, T, O, A, with the calculator sitting in degrees.

Two edges of this worth naming. Every triangle here had a right angle in it, and once that goes, the sine rule and the cosine rule take over. Problems chaining several triangles together, or running in three dimensions, belong to the video on Pythagoras and trigonometry in three dimensions. A circle carries more than these two formulae as well. Arc length and sector area are their own pair, in the video on arcs and sectors, and the angle facts live in the video on circle theorems. Today was one whole circle and one flat triangle.

Next in the chapter: Choosing Between the Quadratic Formula, Sine Rule, Cosine Rule and Area Formula. Four more formulae to hold from memory, and the skill there is picking the right one when the question will not tell you which.

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