Pythagoras' Theorem

Why this chapter matters

Every year the BECE asks at least one question that needs Pythagoras' theorem. It hides inside questions about ladders, football pitches, kites, roofs and the distance between two towns. If you can spot a right-angled triangle, you can answer all of them with one rule.

Kofi is in Basic 9 at a school in Koforidua. His mother wants to know whether a 5-metre ladder will reach the top of their 4-metre wall if its foot stands 2 metres away from it. Kofi does not need to climb anything. He draws a triangle, writes one line of working, and gives her the answer.

By the end of this book you will:

• recognise a right-angled triangle and name its sides,
• state Pythagoras' theorem and explain what it says about squares,
• find the hypotenuse or a shorter side,
• work with square roots, with and without a calculator,
• solve real problems from Ghana and from the exam.

Keep a pencil, an exercise book and (where allowed) a calculator beside you. Do every Try this before turning the page.

The man behind the theorem

Pythagoras lived in Greece about 2,500 years ago. He led a group of thinkers who believed that numbers explained everything, from music to the stars. The rule about right-angled triangles carries his name, but the Babylonians and Egyptians were using it long before he was born. Pythagoras (or his students) may have been the first to prove it.

A proof is different from an example. An example says "it worked for this triangle". A proof says "it must work for every right-angled triangle that could ever be drawn". That is why mathematicians still honour him.

In Ghana, surveyors use the theorem to mark out plots of land, builders use it to check corners are square, and ECG engineers use it to work out the length of a cable that runs diagonally across a field.

What is a right-angled triangle?

A right angle is exactly 90°. It is the angle at the corner of your exercise book, the corner of a door, the angle where a wall meets the floor.

A right-angled triangle is any triangle that has one right angle. The other two angles are both less than 90° and add up to 90°, because the three angles of every triangle add up to 180°.

In diagrams the right angle is marked with a small square in the corner. Always look for that square. If a question says "the ladder leans against a vertical wall on horizontal ground", the wall and the ground meet at a right angle, even if nobody draws the square.

Try this

1. Which of these could be a right-angled triangle? Angles of 90°, 45°, 45°. Angles of 60°, 60°, 60°. Angles of 90°, 30°, 60°.
2. Look around your room. Name three right angles you can see.

Naming the sides

The three sides of a right-angled triangle have names.

• The hypotenuse is the longest side. It is always opposite the right angle. It never touches the right angle.
• The other two sides are the shorter sides, also called the legs. They are the two sides that meet at the right angle.

In most textbooks the hypotenuse is labelled c and the shorter sides a and b. It does not matter which short side you call a and which you call b. It matters very much that c is the hypotenuse.

A quick test: put your finger on the right angle. The two sides touching your finger are the legs. The side far away from your finger is the hypotenuse.

Try this

Draw a right-angled triangle with the right angle at the top left. Label the hypotenuse c and the legs a and b. Then draw another with the right angle at the bottom right and label it again.

The theorem in words

Pythagoras' theorem says:

In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

"The square on a side" means a real square drawn with that side as one edge. If one leg is 3 cm, the square on it has area 3 × 3 = 9 cm². If the other leg is 4 cm, its square has area 16 cm². The theorem promises that the square on the hypotenuse has area 9 + 16 = 25 cm², so the hypotenuse is 5 cm, because 5 × 5 = 25.

Look at the picture. The two smaller squares, cut up and rearranged, fill the big square exactly. Nothing is left over and nothing is missing.

The theorem as a formula

In symbols, with c as the hypotenuse:

a² + b² = c²

Read it as "a squared plus b squared equals c squared". The small 2 means "multiply the number by itself".

The formula only works if c is the hypotenuse. Students lose marks every year by squaring the wrong side. Before you write anything, find the right angle, find the side opposite it, and call that c.

The formula can be rearranged:

• To find the hypotenuse: c² = a² + b²
• To find a shorter side: a² = c² − b² or b² = c² − a²

Notice the pattern. Hypotenuse: add. Shorter side: subtract. The hypotenuse is the biggest side, so its square must be the biggest number.

Squaring a number

To square a number, multiply it by itself. 7² = 7 × 7 = 49. 12² = 144. 1.5² = 2.25.

You should know these square numbers by heart. They appear again and again in BECE questions.

n	n²	n	n²
1	1	8	64
2	4	9	81
3	9	10	100
4	16	11	121
5	25	12	144
6	36	13	169
7	49	15	225

A common mistake is to write 5² = 10. Squaring is not doubling. 5² means 5 × 5, not 5 × 2.

Try this

Work out 14², 20², 2.5² and 0.6² without a calculator. Check your answers with a calculator afterwards if you have one.

Square roots

The square root undoes squaring. Because 5² = 25, the square root of 25 is 5. We write √25 = 5.

Finding the square root is the last step in almost every Pythagoras question, because the formula gives you c², and you want c.

Square roots of square numbers are whole: √49 = 7, √144 = 12. Square roots of other numbers are decimals that never end: √50 = 7.071..., √2 = 1.414...

Estimating without a calculator. √50 lies between √49 = 7 and √64 = 8, and much closer to 7. So √50 ≈ 7.1. If an exam question says "leave your answer in surd form", simply write √50 and stop.

With a calculator. Press the √ key, then the number, then =. On some calculators you type the number first. Try both and see which gives √81 = 9.

Try this

Estimate √30, √70 and √120 to one decimal place. Then check.

Checking the theorem with 3, 4, 5

Before using a rule, it is wise to test it. The simplest test is the 3-4-5 triangle.

3² + 4² = 9 + 16 = 25, and 5² = 25. The rule holds.

Ancient Egyptian builders knew this trick. They took a rope with 12 equal knots and stretched it into a triangle with sides 3, 4 and 5 knots. The angle between the 3 side and the 4 side was a perfect right angle. That is how they set out the square corners of temples and fields.

Builders in Ghana still do the same with a tape measure. Measure 3 feet along one wall, 4 feet along the other; if the diagonal is exactly 5 feet, the corner is square.

Recap 1

Stop and check what you know so far.

• A right-angled triangle has one angle of 90°, marked with a small square.
• The hypotenuse is the longest side and sits opposite the right angle.
• Pythagoras' theorem: a² + b² = c², where c is the hypotenuse.
• Squaring means multiplying a number by itself. Square rooting undoes it.
• 3, 4, 5 is the simplest set of whole numbers that fits the theorem.

Quick quiz (answers at the bottom)

1. In a triangle with sides 6, 8 and 10, which side is the hypotenuse?
2. What is 11²?
3. What is √169?
4. Is 5, 12, 13 a right-angled triangle? Test it.

Answers: 1) 10. 2) 121. 3) 13. 4) Yes, 25 + 144 = 169.

Worked example: finding the hypotenuse

A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.

Step 1. Write the formula. c² = a² + b²

Step 2. Put in the numbers. c² = 6² + 8²

Step 3. Square them. c² = 36 + 64

Step 4. Add. c² = 100

Step 5. Square root. c = √100 = 10 cm

Always finish with the unit. The answer is 10 cm, not just 10.

Notice that 6, 8, 10 is simply 3, 4, 5 doubled. Any multiple of a Pythagorean set (6-8-10, 9-12-15, 30-40-50) is also a Pythagorean set. Spotting this saves time, but always show your working in the exam. The marks are for the method as much as the answer.

Try this: the hypotenuse

Find the hypotenuse in each triangle. Show all five steps and round to one decimal place where needed.

1. Shorter sides 5 cm and 12 cm.
2. Shorter sides 9 m and 12 m.
3. Shorter sides 7 cm and 24 cm.
4. Shorter sides 4 cm and 5 cm.
5. Shorter sides 2 m and 3 m.

Before you look at the answers, ask yourself: is my hypotenuse longer than both shorter sides? If not, something is wrong.

Answers: 1) 13 cm. 2) 15 m. 3) 25 cm. 4) √41 ≈ 6.4 cm. 5) √13 ≈ 3.6 m.

Questions 1 to 3 give whole numbers. Questions 4 and 5 do not; that is normal. Most real measurements give messy square roots, and the exam will tell you how many decimal places to use.

Worked example: finding a shorter side

A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the other side.

This time we subtract, because we are looking for a leg, not the hypotenuse.

Step 1. b² = c² − a²

Step 2. b² = 13² − 5²

Step 3. b² = 169 − 25

Step 4. b² = 144

Step 5. b = √144 = 12 cm

The most common error is to add here and get √194 ≈ 13.9 cm. That cannot be right, because a shorter side cannot be longer than the hypotenuse. Checking your answer against common sense catches this every time.

Memory tip: the biggest number always stands alone on one side of the equals sign. c² = a² + b². If you know c, move things so that c² is by itself and the other square is subtracted.

Try this: a shorter side

Find the missing shorter side. Round to one decimal place where necessary.

1. Hypotenuse 10 cm, one leg 6 cm.
2. Hypotenuse 25 m, one leg 7 m.
3. Hypotenuse 17 cm, one leg 8 cm.
4. Hypotenuse 9 cm, one leg 4 cm.
5. Hypotenuse 6.5 m, one leg 2.5 m.

Checklist for every answer:

• Did I identify the hypotenuse correctly?
• Did I subtract, not add?
• Is my answer shorter than the hypotenuse?
• Did I write the unit?

Answers: 1) 8 cm. 2) 24 m. 3) 15 cm. 4) √65 ≈ 8.1 cm. 5) 6 m.

Is it right-angled? The converse

The theorem works in reverse too. If the three sides of a triangle satisfy a² + b² = c², where c is the longest side, then the triangle must have a right angle opposite c. This is called the converse of Pythagoras' theorem.

Example. Abena measures a triangular garden bed: 8 m, 15 m and 17 m. Is the corner between the 8 m and 15 m sides a right angle?

Longest side is 17. 17² = 289.

8² + 15² = 64 + 225 = 289.

They match, so yes, it is a right angle.

Example. Sides 5, 6, 8. Longest is 8, and 8² = 64. But 5² + 6² = 25 + 36 = 61. Not equal, so the triangle is not right-angled.

Try this

Test these: (9, 12, 15), (7, 9, 11), (20, 21, 29), (10, 24, 26).

A proof you can see

Why is the theorem true? Here is an old Chinese proof that needs no algebra.

Take a right-angled triangle with legs a and b and hypotenuse c. Make four copies. Arrange them inside a big square of side a + b, as in the picture, so that their hypotenuses form a tilted square in the middle.

The area of the big square can be counted in two ways.

• As one square: (a + b)² = a² + 2ab + b².
• As four triangles plus the tilted square: 4 × ½ab + c² = 2ab + c².

These must be equal, so a² + 2ab + b² = 2ab + c². Take away 2ab from both sides and you are left with a² + b² = c².

You do not need to reproduce this in the BECE, but understanding it helps the rule stick.

Many proofs, one theorem

Mathematicians have found more than 350 different proofs of Pythagoras' theorem. One was written by an American president, James Garfield, while he was still a congressman. Another uses similar triangles. Another cuts the squares into pieces that slide into place.

Why so many? Because the theorem is at the centre of geometry. It links lengths to areas, and later it links to trigonometry, to coordinates and to the distance formula you will meet in SHS.

The video below shows some of the cleverest proofs in a few minutes.

Recap 2

• Hypotenuse unknown: c² = a² + b². Add, then square root.
• Shorter side unknown: a² = c² − b². Subtract, then square root.
• Converse: if a² + b² = c² for the longest side c, the triangle is right-angled.
• Always check: the hypotenuse is the longest side; a leg is shorter than the hypotenuse.
• Always write the unit.

Quick quiz

1. Legs 12 and 16. Hypotenuse?
2. Hypotenuse 26, leg 10. Other leg?
3. Is (6, 7, 9) right-angled?
4. What is √(7² + 24²)?

Answers: 1) 20. 2) 24. 3) No, 36 + 49 = 85, not 81. 4) 25.

Real problem: the ladder

Now the part that matters most: turning a story into a triangle.

Problem. A 5 m ladder leans against a vertical wall. Its foot is 2 m from the base of the wall. How high up the wall does the ladder reach?

Draw it. The wall is vertical, the ground is horizontal, so they meet at a right angle. The ladder is the slanted side, so the ladder is the hypotenuse. The height up the wall is a shorter side.

Solve it. h² = 5² − 2² = 25 − 4 = 21. h = √21 ≈ 4.6 m (1 d.p.).

So Kofi's mother's ladder from page one does reach a 4-metre wall, with a little to spare.

Ladder questions almost always make the ladder the hypotenuse. Draw first, label second, calculate third.

Try this: ladders and poles

1. A ladder 6.5 m long reaches 6 m up a wall. How far is its foot from the wall?
2. A painter needs to reach a window 3.6 m high. His ladder's foot must stay 1.5 m from the wall for safety. How long must the ladder be?
3. A flagpole at Kwame's school is held by a wire fixed to the top of the pole and to the ground 4 m from its base. The wire is 9 m long. How tall is the pole? (1 d.p.)
4. A 4 m ladder leans with its foot 3 m from the wall. Explain why this is a dangerous way to use it, using your answer for the height.

Answers: 1) 2.5 m. 2) 3.9 m. 3) 8.1 m. 4) It reaches only √7 ≈ 2.6 m, and at that angle it is likely to slip.

Real problem: the football pitch

Problem. A full-size football pitch is 105 m long and 68 m wide. Ama runs diagonally from one corner flag to the opposite one. How far does she run?

The length and the width meet at a right angle at the corner flag. The diagonal is the hypotenuse.

d² = 105² + 68² = 11,025 + 4,624 = 15,649

d = √15,649 ≈ 125.1 m

Compare: running along two sides would be 105 + 68 = 173 m. The diagonal saves her almost 48 m. That is why footballers, and goats crossing a field, take the diagonal.

Try this

The school's smaller pitch at Tamale measures 90 m by 45 m. How long is its diagonal? (1 d.p.)

Real problem: rectangles and diagonals

Any rectangle can be split into two right-angled triangles by its diagonal. That makes Pythagoras useful for doors, television screens, plots of land and boxes.

Example. A rectangular plot in Kasoa is 40 m by 30 m. The owner wants to fence it and also run a path along one diagonal. How long is the path?

d² = 40² + 30² = 1,600 + 900 = 2,500, so d = 50 m. (Notice 3-4-5 scaled by 10.)

Example. A television is sold as "32 inch". That is the diagonal. If the screen is 28 inches wide, how tall is it?

h² = 32² − 28² = 1,024 − 784 = 240, so h = √240 ≈ 15.5 inches.

Try this

1. A door is 2 m high and 0.8 m wide. Can a 2.2 m board fit through it diagonally?
2. A square has a diagonal of 10 cm. Find its side length. (Hint: both legs are equal.)

Real problem: distances on a map

Pythagoras finds straight-line distances when you know how far east and how far north something is.

Problem. From Kofi's house, the market is 600 m east. From the market, the clinic is 800 m north. How far is the clinic from Kofi's house as the crow flies?

East and north are at right angles. So:

d² = 600² + 800² = 360,000 + 640,000 = 1,000,000

d = √1,000,000 = 1,000 m = 1 km

Problem. A bus travels 24 km due east from Sunyani, then 10 km due north. How far is it from Sunyani?

d² = 24² + 10² = 576 + 100 = 676, so d = 26 km.

Compass directions (north, south, east, west) always give you right angles. When a question mixes them, draw a quick sketch with north at the top.

Try this: journeys

Sketch each journey, mark the right angle, then calculate. Round to one decimal place.

1. A fishing boat leaves Elmina and sails 5 km south, then 12 km west. How far is it from the harbour?
2. Abena walks 300 m north to school, then 400 m east to the library. How far is the library from her home?
3. A plane flies 150 km east and then 200 km north. How far is it from where it started?
4. A drone flies 2.4 km due west and is then 2.6 km from its base in a straight line. How far north or south did it also move?

Answers: 1) 13 km. 2) 500 m. 3) 250 km. 4) 1 km.

Recap 3

Turning words into triangles:

• Ladder / pole / wire against a wall: the leaning object is the hypotenuse.
• Rectangle / pitch / plot / screen: the diagonal is the hypotenuse.
• Journey with compass directions: the straight-line distance is the hypotenuse.

The method never changes:

1. Draw and label.
2. Decide: hypotenuse (add) or shorter side (subtract)?
3. Square, add or subtract, square root.
4. Round as instructed. Write the unit.

Quick quiz

1. A 10 m ladder, foot 6 m from the wall. Height?
2. Rectangle 12 cm by 5 cm. Diagonal?
3. 9 km east then 12 km south. Distance from start?

Answers: 1) 8 m. 2) 13 cm. 3) 15 km.

Working with decimals

Exam measurements are not always whole numbers. The method is the same; you just need care with the arithmetic.

Example. Legs 1.5 m and 2 m.

c² = 1.5² + 2² = 2.25 + 4 = 6.25, so c = √6.25 = 2.5 m.

Example. Hypotenuse 7.5 cm, leg 4.5 cm.

b² = 7.5² − 4.5² = 56.25 − 20.25 = 36, so b = 6 cm.

Both of these are 3-4-5 triangles in disguise (multiplied by 0.5 and 1.5). Many exam questions are.

Tips for decimals

• Square carefully: 1.5² = 2.25, not 2.5 and not 3.
• Keep all the decimal places until the very last step. Round only your final answer.
• If the answer is "4.58257...", and the question says 2 d.p., write 4.58.

Try this

Legs 0.9 m and 1.2 m. Hypotenuse? Hypotenuse 3.4 m, leg 1.6 m. Other leg?

Leaving answers in surd form

A surd is a square root that cannot be simplified to a whole number, like √5 or √50.

Sometimes a question says "give your answer in surd form" or "leave your answer as a square root". Then you stop before pressing the √ button.

Example. Legs 3 cm and 5 cm. c² = 9 + 25 = 34, so c = √34 cm. Done.

Some surds can be simplified. Look for a square-number factor.

√50 = √(25 × 2) = √25 × √2 = 5√2

√72 = √(36 × 2) = 6√2

√48 = √(16 × 3) = 4√3

Try this

Simplify √18, √80, √200 and √45. Then give the hypotenuse of a right-angled triangle with legs 6 and 6 in simplest surd form.

Answers: 3√2, 4√5, 10√2, 3√5; √72 = 6√2.

Pythagorean triples

A Pythagorean triple is three whole numbers that fit a² + b² = c².

The ones worth memorising:

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25
• 20, 21, 29

And all their multiples: 6-8-10, 9-12-15, 12-16-20, 15-20-25, 10-24-26, 16-30-34...

Why memorise them? Because examiners love them. When you see a hypotenuse of 13 and a leg of 5, you can write "12" with confidence, then show the working to earn the marks.

Try this

1. Which triple is hiding in a triangle with legs 21 and 28?
2. Which triple is hiding in hypotenuse 51, leg 24?
3. Make up a new triple by multiplying 7, 24, 25 by 3.

Answers: 1) 3-4-5 × 7, hypotenuse 35. 2) 8-15-17 × 3, other leg 45. 3) 21, 72, 75.

Isosceles triangles and heights

A triangle that is not right-angled can often be cut into two right-angled triangles. Then Pythagoras works.

Example. An isosceles triangle has two equal sides of 10 cm and a base of 12 cm. Find its height.

Draw the height from the top vertex down to the middle of the base. It splits the base into two halves of 6 cm each and makes two right-angled triangles. In one of them, the hypotenuse is 10 cm and one leg is 6 cm.

h² = 10² − 6² = 100 − 36 = 64, so h = 8 cm.

Now you can even find the area: ½ × 12 × 8 = 48 cm².

Try this

An equilateral triangle has sides of 10 cm. Cut it in half and find its height to one decimal place. (Hint: the half base is 5 cm.)

Answer: √75 ≈ 8.7 cm.

Recap 4

• Decimals: square carefully, round only at the end.
• Surds: stop at √34 if asked; simplify √50 to 5√2 by finding square factors.
• Triples to know: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.
• Isosceles and equilateral triangles split into two right-angled triangles along the height.

Quick quiz

1. Legs 2.4 and 1. Hypotenuse?
2. Simplify √128.
3. Legs 10 and 24: which triple?
4. Isosceles triangle, equal sides 13, base 10. Height?

Answers: 1) 2.6. 2) 8√2. 3) 5-12-13 × 2, hypotenuse 26. 4) 12.

Worked BECE-style question 1

A ladder 7 m long leans against a vertical wall with its foot on horizontal ground 2.5 m from the wall. Calculate, correct to one decimal place, how high up the wall the ladder reaches.

Working

Let the height be h.

h² + 2.5² = 7² (the ladder is the hypotenuse)

h² = 49 − 6.25

h² = 42.75

h = √42.75 = 6.538...

h ≈ 6.5 m

How the marks are given. Usually one mark for the correct equation, one for the correct value of h², and one for the final rounded answer with unit. Skipping steps risks losing the method marks even when your answer is right.

Write neatly, one step per line, with the equals signs lined up.

Worked BECE-style question 2

The diagonal of a rectangular field is 85 m and its length is 84 m. Find (a) the width of the field, (b) the perimeter of the field.

(a) The diagonal is the hypotenuse.

w² = 85² − 84² = 7,225 − 7,056 = 169

w = √169 = 13 m

(b) Perimeter = 2 × (length + width) = 2 × (84 + 13) = 2 × 97 = 194 m

Notice how the question had two parts. Part (b) depended on part (a). If you had made an error in (a), examiners still give marks in (b) for using your own answer correctly, so never leave (b) blank.

Try this

A rectangular hall has width 9 m and diagonal 41 m. Find its length and its area.

Answer: length 40 m, area 360 m².

Worked BECE-style question 3

A boy walks 300 m due east and then 400 m due north. (a) Draw a sketch. (b) Find how far he is from his starting point. (c) He returns to the start by the shortest path. Find the total distance walked.

(a) Sketch a horizontal arrow 300 m to the right, then a vertical arrow 400 m up. Join the end back to the start with a dotted line. Mark the right angle.

(b) d² = 300² + 400² = 90,000 + 160,000 = 250,000, so d = 500 m.

(c) Total = 300 + 400 + 500 = 1,200 m or 1.2 km.

Sketches earn marks. They must have the right angle marked and the lengths labelled, but they do not need to be to scale.

Try this: exam practice

Answer in full, with working.

1. A wire 25 m long runs from the top of a telecommunication mast to a point on the ground 7 m from the mast's base. How tall is the mast?
2. A rectangular garden is 18 m by 24 m. Find the length of its diagonal.
3. A kite string is 50 m long. The kite is directly above a point 30 m from Abena. How high is the kite?
4. From Cape Coast, a boat sails 16 km south then 30 km east. How far is it from Cape Coast?
5. A triangle has sides 11 cm, 60 cm and 61 cm. Show whether it is right-angled.
6. A square room has a diagonal of 7 m. Find the length of one side to two decimal places.

Answers: 1) 24 m. 2) 30 m. 3) 40 m. 4) 34 km. 5) Yes, 121 + 3,600 = 3,721. 6) 4.95 m.

Common mistakes to avoid

Mistake 1: squaring the wrong side. The hypotenuse must be c. Find the right angle first.

Mistake 2: adding when you should subtract. If the hypotenuse is known, subtract.

Mistake 3: forgetting the square root. c² = 169 is not the answer; c = 13 is.

Mistake 4: doubling instead of squaring. 9² is 81, not 18.

Mistake 5: rounding too early. Keep full decimals until the final line.

Mistake 6: no unit. 13 what? Centimetres, metres, kilometres?

Mistake 7: answers that fail common sense. A leg longer than the hypotenuse, or a ladder that reaches higher than its own length, is a sign to go back and check.

Try this

A student wrote: "Hypotenuse 10, leg 8. Other leg: 10² + 8² = 164, √164 = 12.8." Find the mistake and give the correct answer.

Answer: should subtract. 100 − 64 = 36, other leg 6.

Recap 5

The full exam method, in one place:

1. Read the question twice. Underline the numbers.
2. Sketch the triangle. Mark the right angle and label the sides.
3. Identify the hypotenuse: opposite the right angle, the longest.
4. Write the formula: c² = a² + b² (or rearranged).
5. Substitute and calculate step by step.
6. Square root.
7. Round as instructed and add the unit.
8. Check that the answer makes sense.

Pythagoras in three dimensions

The theorem can be used twice to find the longest diagonal inside a box, from one bottom corner to the opposite top corner.

Example. A box measures 12 cm long, 9 cm wide and 8 cm high.

First find the diagonal of the base: d₁² = 12² + 9² = 144 + 81 = 225, so d₁ = 15 cm.

Now that base diagonal and the height form a new right-angled triangle. d₂² = 15² + 8² = 225 + 64 = 289, so d₂ = 17 cm.

In fact you can combine the two steps: d² = 12² + 9² + 8² = 289. The longest diagonal of any box is √(length² + width² + height²).

Try this

Can a 1 m straight rod fit inside a box that is 80 cm by 50 cm by 40 cm? Work out the longest diagonal and decide.

Answer: √(6400 + 2500 + 1600) = √10,500 ≈ 102.5 cm, so yes, just.

Pythagoras and similar triangles

In a right-angled triangle, draw the height from the right angle to the hypotenuse. This splits the big triangle into two smaller triangles. All three triangles are similar: the same shape, different sizes.

This is another way to prove the theorem, and it is the bridge to trigonometry, which you will study in SHS. Sine, cosine and tangent are simply ratios of sides in right-angled triangles, and Pythagoras gives the identity sin²θ + cos²θ = 1.

For now, it is enough to know that the theorem is not an isolated trick. It connects to similar shapes, to coordinates, to circles and to almost every topic in SHS mathematics.

Pythagoras on a grid

On squared paper, the distance between two points can be found by drawing a right-angled triangle.

Example. Point A is at (1, 2) and point B is at (7, 10). How far apart are they?

Going from A to B: 6 units across (7 − 1) and 8 units up (10 − 2). Those are the legs.

d² = 6² + 8² = 36 + 64 = 100, so d = 10 units.

This is the distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²). It is just Pythagoras written for coordinates.

Try this

Find the distance between:

1. (0, 0) and (5, 12)
2. (2, 3) and (10, 9)
3. (−3, 1) and (4, 1). (Careful: this one is horizontal. What is the vertical leg?)

Answers: 1) 13. 2) 10. 3) 7, no triangle needed.

Checking with construction

You can test the theorem with a ruler and a pair of compasses.

1. Draw a horizontal line 8 cm long. Call its ends A and B.
2. At B, use a set square or protractor to draw a line going straight up at 90°.
3. Mark point C on it, 6 cm above B.
4. Join A to C and measure AC.

Pythagoras predicts AC = √(64 + 36) = √100 = 10 cm. If you drew carefully, your ruler will say 10 cm, or very close. If you get 9.8 or 10.2, your right angle or your lengths were slightly off. That is normal; drawing is never perfect, but the theorem is.

Try this

Repeat with AB = 12 cm and BC = 5 cm. Predict AC first, then measure.

Pythagoras at work in Ghana

Surveying. Before a plot can be registered with the Lands Commission, a surveyor measures it. Diagonals are checked with Pythagoras to be sure the corners are true right angles.

Building. Masons use 3-4-5 (or 6-8-10 in feet) to square the foundation of a house before the first block is laid.

Electricity. GRIDCo and ECG engineers calculate cable lengths for lines that cross valleys diagonally rather than following roads.

Carpentry. A roof truss is a set of triangles. The slanted rafter is a hypotenuse; its length comes from the span and the rise.

Sport. Groundsmen mark pitches by measuring the diagonal, then adjust until it matches the calculated value.

Try this

A roof has a span of 8 m (so each half is 4 m) and rises 3 m at the centre. How long is each rafter?

Answer: 5 m.

Recap 6

• 3D boxes: longest diagonal = √(l² + w² + h²).
• Coordinates: distance = √((Δx)² + (Δy)²).
• Similar triangles inside a right-angled triangle lead to trigonometry.
• Construction lets you test the theorem with a ruler.
• The theorem is used daily by surveyors, builders, engineers and groundsmen.

Quick quiz

1. Box 3 cm × 4 cm × 12 cm. Longest diagonal?
2. Distance from (1, 1) to (4, 5)?
3. A rafter spans a 6 m half-width and rises 2.5 m. Length?

Answers: 1) 13 cm. 2) 5. 3) 6.5 m.

Mixed practice

Choose the method, then solve. Round to one decimal place where needed.

1. Legs 15 and 20. Hypotenuse?
2. Hypotenuse 29, leg 21. Other leg?
3. Is (12, 35, 37) right-angled?
4. A ladder 8 m long rests with its top 7 m up a wall. How far is its foot from the wall?
5. A rectangular carpet is 3 m by 4 m. Find the diagonal.
6. Simplify √98.
7. A boat sails 7 km north and 24 km east. Distance from start?
8. Equilateral triangle of side 12 cm. Height?
9. A box 6 cm × 8 cm × 24 cm. Longest diagonal?
10. Points (2, −1) and (8, 7). Distance?

Answers: 1) 25. 2) 20. 3) Yes. 4) 3.9 m. 5) 5 m. 6) 7√2. 7) 25 km. 8) 10.4 cm. 9) 26 cm. 10) 10.

Challenge problems

These go a little beyond the basic exam, for students who want to stretch.

1. A square has an area of 50 cm². Find the length of its diagonal exactly.
2. Two poles, 5 m and 11 m tall, stand 8 m apart on level ground. A wire joins their tops. How long is the wire?
3. A ladder leans against a wall reaching 12 m. It is pulled 4 m further from the wall and now reaches only 9 m. How long is the ladder? (Hint: let the first distance be x, and write two equations.)
4. Find three different Pythagorean triples that have 60 as one of the legs.

Answers: 1) 10 cm. 2) 10 m. 3) 15 m. 4) For example 11-60-61, 60-80-100 (3-4-5 × 20), 25-60-65 (5-12-13 × 5).

If question 3 defeats you, that is fine; it uses simultaneous equations, which you will meet properly in SHS.

Revision card

Copy this onto a small card and keep it in your maths book.

Pythagoras' theorem — right-angled triangles only.

a² + b² = c² (c = hypotenuse = opposite the right angle = longest)

• Find hypotenuse: c = √(a² + b²)
• Find a leg: a = √(c² − b²)
• Right-angled? Test: does longest² = sum of other two squares?

Triples: 3-4-5 · 5-12-13 · 8-15-17 · 7-24-25 · 20-21-29

Surds: √50 = 5√2 · √72 = 6√2 · √48 = 4√3

3D: d = √(l² + w² + h²)

Grid: d = √((Δx)² + (Δy)²)

Always: sketch · label · formula · substitute · square root · round · unit · check.

Learn more

• Maths is Fun: Pythagoras' Theorem — a clear page with interactive examples and a quiz.
• BBC Bitesize: Pythagoras' theorem — short lessons and tests at the right level for JHS.
• Khan Academy: The Pythagorean theorem — videos and practice questions with instant marking.
• NaCCA — Ghana's curriculum body, where the Basic 9 maths syllabus lives.

Watch more

Think about it

1. Why does the theorem only work for right-angled triangles? What would happen to c² if the angle between a and b were bigger than 90°?
2. A builder has no set square, only a tape measure. Explain, step by step, how she can make a perfect right-angled corner.
3. Why do you think so many exam questions use 3-4-5 and 5-12-13 triangles?
4. The diagonal of a square with side 1 is √2, a number that can never be written exactly as a fraction. Why might that have troubled Pythagoras and his followers?
5. Where in your own town could Pythagoras' theorem save someone time, money or materials?