Pythagoras: worked examples and method

The rule

a² + b² = c², where c is the hypotenuse.

The right angle is between the two shorter sides.

Find the hypotenuse

Square both legs, add, then square-root.

3² + 4² = 9 + 16 = 25, and √25 = 5.

Find a shorter side

Subtract the known square from the hypotenuse squared, then square-root.

c² − a² = b². Do not add when the hypotenuse is already known.

Only right-angled triangles

If there is no right angle, Pythagoras does not apply.

A different triangle needs the sine or cosine rule later.

A diagram

Mark the right angle, then label the side opposite it as c.

A sloping side is not automatically the hypotenuse unless it faces the right angle.

Exact squares

3, 4, 5 and their multiples are exact.

5, 12, 13 and 8, 15, 17 are other useful triples.

Finding the hypotenuse

Start by marking the right angle. The hypotenuse is the side facing that mark. In the triangle here the shorter sides are 8 cm and 15 cm, so the hypotenuse is the sloping side labelled c. It is not one of the sides that form the square corner.

Square each shorter side, add those squares, and square-root the total. For these lengths, 8^2 + 15^2 = 64 + 225 = 289, and √(289) = 17. So the hypotenuse is 17 cm. Round only at the end, to the accuracy the question asks for.

Method: 1. Mark the right angle and label the opposite side c. 2. Square each shorter side and add the squares. 3. Square-root that total to get c. 4. Round only at the end, and check that c is the longest side.

Tip: A quick check for a whole-number answer is a Pythagorean triple, such as 3, 4, 5 or 8, 15, 17. If your roots are messy, the triangle may simply not be one of those.

Finding a shorter side

Sometimes the hypotenuse is the side you are given. A ladder, a diagonal of a rectangle, or the longest side of a right-angled triangle drawn on the page is often that length. You still start from a^2 + b^2 = c^2, but you rearrange it before you calculate.

The square of the hypotenuse is the total of the two smaller squares. Take away the square you already know: a^2 = c^2 - b^2. Then square-root. The answer must come out shorter than c. If it does not, you added instead of subtracting.

Method: 1. Decide which given length is the hypotenuse. 2. Write a^2 = c^2 - b^2. 3. Work out the two squares and subtract. 4. Square-root, then check the side is shorter than c.

Tip: Say the rearrangement in words before you type it: "hypotenuse squared, minus the other square".

Is the triangle right-angled?

You can run the theorem backwards. If the squares of the two shorter sides add up to the square of the longest side, the triangle is right-angled, and the right angle sits opposite the longest side. If they do not match, it is not right-angled, and Pythagoras does not find a missing side.

Take sides 9 cm, 10 cm and 12 cm. The longest side is 12 cm, so test 9^2 + 10^2 against 12^2. That is 81 + 100 = 181, and 12^2 = 144. Because 181 ≠ 144, the triangle is not right-angled.

Method: 1. Pick out the longest side. 2. Square the other two sides and add them. 3. Square the longest side. 4. If the two results are equal, it is right-angled. If not, it is not.

Distance between two points

Two points on a coordinate grid can be joined by a slanted line. That line is the hypotenuse of a right-angled triangle. One leg is the horizontal gap, the change in x. The other leg is the vertical gap, the change in y.

A negative coordinate is allowed. From x = -2 to x = 2 the gap is 2 - (-2) = 4. Squaring makes the sign disappear, so the distance d = √((x_2 - x_1)^2 + (y_2 - y_1)^2) is positive.

Method: 1. Subtract the x-coordinates. Subtract the y-coordinates. 2. Square both gaps and add them. 3. Square-root the total. That is the distance. 4. A negative gap is fine: its square is positive.

Tip: If both points sit on a horizontal or vertical line, one gap is 0 and the distance is just the other gap. You do not need the formula.

A diagonal in 3D (Higher)

A cuboid has three edges meeting at a corner: length, width and height. The straight line through the middle, from one corner to the opposite corner, is longer than any face diagonal. Call it L. This is Higher work. Foundation questions stay on flat right-angled triangles.

First find the base diagonal from the length and the width, d^2 = l^2 + w^2. Then bring in the height: L^2 = d^2 + h^2. Together that is L = √(l^2 + w^2 + h^2). Keep d^2 exact. Do not round a face diagonal before you add the height.

Method: 1. Square the length, the width and the height. 2. Add the three squares. Do not round yet. 3. Square-root the total to get the space diagonal. 4. Check it is longer than each edge and longer than a face diagonal.

Tip: If you only have two dimensions in the square root, you have found a face diagonal, not the corner-to-corner length.

Finding the right-angled triangle inside a shape

In many exam questions there is no obvious right-angled triangle. You have to make one. Drop a perpendicular from the top of an isosceles triangle to its base and you create two right-angled triangles with a base that is half the original. Draw a diagonal across a rectangle and you get a right angle at every corner.

The same idea works for a trapezium with two right angles, a rhombus (the diagonals cross at right angles and bisect each other) and a chord in a circle (the line from the centre to the middle of a chord is perpendicular to it). Sketch the new triangle separately and label only the sides you know.

Method: 1. Draw the shape and add the line that creates a right angle. 2. Redraw the right-angled triangle on its own and label the known sides. 3. Decide whether the unknown side is the hypotenuse or a shorter side. 4. Use a^2 + b^2 = c^2 or a^2 = c^2 - b^2, then answer the original question.

Tip: In an isosceles triangle the perpendicular height always splits the base in half. Use the half, not the whole base.

Two-stage problems and keeping accuracy

Higher-scoring questions use Pythagoras twice. You find one length, then use it as a side in a second triangle. The best way to keep full accuracy is to keep the square of the first length instead of rounding its square root. For example, if AC^2 = 40 then use 40 directly when you work out AD^2.

Only round the final answer, and say the number of decimal places or significant figures in your working. If the question asks for a decimal, give one. If a length comes out as a whole number such as 15, check that it is a sensible size for the sketch before you move on.

Method: 1. Find the square of the first unknown side and keep it unrounded. 2. Use that square in the second use of Pythagoras. 3. Take the square root only at the end. 4. Round to the number of decimal places asked for and add the unit.

Exact answers in surd form (Higher)

Some non-calculator questions say "give your answer as a surd" or "show that the length is 6√(5) cm". Do not turn the square root into a decimal. Find the sum or difference of the squares, then simplify by taking out the largest square number. For √(180), write 180 = 36 × 5, so √(180) = 6√(5).

In a "show that" question every line of working is a mark, so write a^2 + b^2 with numbers, the total under the root, the factorisation and the final surd. The answer is already given, so the marks are for the method and the clear steps, not for a final decimal. Practise spotting square factors such as 4, 9, 16, 25 and 36, because the quickest simplification uses the largest one.

Method: 1. Square each side and add or subtract as the question needs. 2. Write the answer as the square root of that total. 3. Factorise the number under the root using the largest square factor. 4. Write the simplified surd and compare it with the form given in the question.

How exam questions are set out

A one-triangle question usually says "Work out the value of x" or "Use Pythagoras' theorem to work out the length" with the note "Not drawn accurately". The tariff is usually 3 marks: one for squaring and adding or subtracting, one for the square root, and one for the correct answer with the rounding asked for.

Decision questions such as "Is the triangle right-angled?" or "Is the ladder long enough?" end with a conclusion in words. Always compare two numbers: a^2 + b^2 against c^2, or the length you worked out against the length you need. A bare "Yes" with no evidence earns very little. A written conclusion takes ten seconds and is often worth the final mark.

Method: 1. Identify the hypotenuse: it is opposite the right angle and is the longest side. 2. Write the full equation with numbers before you press any keys. 3. Check the hypotenuse is longer than both shorter sides. 4. In a yes or no question, finish with a sentence that uses your two numbers.

Tip: If your answer for a shorter side is bigger than the hypotenuse, you added when you should have subtracted.