What is Pythagoras Theorem and how is it used in Geometry?

Pythagoras Theorem is a fundamental principle in Geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as a² + b² = c². It is used to calculate the length of one side of a right-angled triangle when the lengths of the other two sides are known, making it a crucial tool in solving various geometric problems.

How can I apply Pythagoras Theorem to solve problems in the Edexcel IGCSE Mathematics exam?

In the Edexcel IGCSE Mathematics exam, Pythagoras Theorem can be applied to solve problems involving right-angled triangles. To use the theorem, identify the right angle and label the sides as 'a', 'b', and 'c', where 'c' is the hypotenuse. Substitute the known values into the equation a² + b² = c² and solve for the unknown side. This method is often used in questions requiring the calculation of distances or heights.

What are some common mistakes to avoid when using Pythagoras Theorem?

Common mistakes include misidentifying the hypotenuse, which is always the longest side opposite the right angle, and incorrectly squaring the side lengths. Another error is forgetting to take the square root after summing the squares of the two shorter sides. Always double-check your calculations and ensure that the triangle is right-angled before applying the theorem.

Can Pythagoras Theorem be used in non-right-angled triangles?

No, Pythagoras Theorem is specifically applicable only to right-angled triangles. For non-right-angled triangles, other methods such as the Law of Sines or the Law of Cosines are used to find unknown sides or angles. These laws extend the principles of trigonometry to all types of triangles.

How does Pythagoras Theorem relate to Trigonometry in the Edexcel IGCSE curriculum?

Pythagoras Theorem is closely related to Trigonometry, as both deal with the properties of triangles. In the Edexcel IGCSE curriculum, understanding Pythagoras Theorem provides a foundation for learning trigonometric ratios such as sine, cosine, and tangent, which are used to solve problems involving angles and lengths in right-angled triangles. Mastery of the theorem aids in comprehending more advanced trigonometric concepts.

Why is "Using $a^{2} + b^{2} = c^{2}$ when finding a leg" a common mistake in Pythagoras Theorem?

Why it happens: Memorising one direction. How to avoid it: If finding the HYPOTENUSE: c^2 = a^2 + b^2. If finding a LEG: a^2 = c^2 - b^2.

Why is "Applying Pythagoras to non-right triangles" a common mistake in Pythagoras Theorem?

Why it happens: Forgetting condition. How to avoid it: Pythagoras requires a RIGHT ANGLE. For other triangles, use cosine rule.

Why is "Forgetting to take square root at the end" a common mistake in Pythagoras Theorem?

Why it happens: Stopping after computing c^2. How to avoid it: Pythagoras gives c^2. Square root to get c.

Geometry and TrigonometryEdexcel IGCSE Mathematics (4MA1)

Pythagoras Theorem

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Pythagoras' Theorem Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

$a^{2} + b^{2} = c^{2}$ . Find missing sides of right triangles, distances between points, diagonals.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

4.7 — Use Pythagoras' theorem in 2D problems.
4.7 — Use Pythagoras in 3D contexts.
4.7 — Find distance between two points using Pythagoras.

2D Pythagoras

$a^{2} + b^{2} = c^{2}$ . Identify hypotenuse first.

Theorem: In a right-angled triangle:

$a^{2} + b^{2} = c^{2}$

where $c$ is the HYPOTENUSE (longest side, opposite the right angle).

To find the hypotenuse: $c = \sqrt{a^{2} + b^{2}}$ .

To find a leg: $a = \sqrt{c^{2} - b^{2}}$ .

Worked example. Right triangle, legs $3$ and $4$ .

$c^{2} = 9 + 16 = 25$ , so $c = 5$ .

Worked example. Hypotenuse $13$ , one leg $5$ .

$5^{2} + b^{2} = 169 \to b^{2} = 144 \to b = 12$ .

Common Pythagorean triples (memorise — saves time):

Triple	$a^{2} + b^{2}$
$3, 4, 5$	$9 + 16 = 25$
$5, 12, 13$	$25 + 144 = 169$
$8, 15, 17$	$64 + 225 = 289$
$7, 24, 25$	$49 + 576 = 625$

Also multiples: $6, 8, 10$ (= $2 \times 3, 4, 5$ ); $9, 12, 15$ ; etc.

Worked qualitative. Spotting a triple saves arithmetic.

$5, 12, ?$ → recognise as $5, 12, 13$ .
$9, 12, ?$ → recognise as $3, 4, 5$ scaled by $3$ → $15$ .

Edexcel tip. Identify HYPOTENUSE first. The largest side. Then decide if you're finding it or a leg.

$c$ = hypotenuse (longest, opposite right angle).
Hypotenuse: $\sqrt{a^{2} + b^{2}}$ .
Leg: $\sqrt{c^{2} - b^{2}}$ .
Memorise common triples.

3D Pythagoras

$d^{2} = a^{2} + b^{2} + c^{2}$ for cuboid diagonal.

3D Pythagoras finds the long diagonal of a cuboid (or the longest distance in 3D space).

For cuboid with sides $a, b, c$ :

$d^{2} = a^{2} + b^{2} + c^{2}$

Why? Apply 2D Pythagoras twice.

Floor diagonal: $d_{\text{floor}}^{2} = a^{2} + b^{2}$ .
Long diagonal: $d^{2} = d_{\text{floor}}^{2} + c^{2} = a^{2} + b^{2} + c^{2}$ .

Worked example. Cuboid $4 \times 6 \times 12$ .

$d^{2} = 16 + 36 + 144 = 196$ .
$d = 14$ .

Other 3D applications:

Distance from corner to opposite corner of a room.
Distance from vertex of a pyramid to a corner of the base.
Length of a string in a 3D structure.

Worked qualitative. A box is $3 \times 4 \times 12$ inside. Will a $13$ -inch ruler fit diagonally?

Diagonal: $\sqrt{9 + 16 + 144} = \sqrt{169} = 13$ .
Ruler fits exactly along the long diagonal. Just.

Edexcel tip. State the formula explicitly. Mark schemes credit recognising 3D Pythagoras as separate from 2D.

$d^{2} = a^{2} + b^{2} + c^{2}$ .
Cuboid diagonals.
Apply 2D Pythagoras twice = same result.
Useful for 3D distance problems.

Distance between two points

$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$ .

Distance formula is just Pythagoras applied to coordinates.

For points $A(x_1, y_1)$ and $B(x_2, y_2)$ :

$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$

Why? Form a right triangle:

Horizontal leg: $|x_2 - x_1|$ .
Vertical leg: $|y_2 - y_1|$ .
Hypotenuse: $d$ .
Pythagoras: $d^{2} = (\Delta x)^{2} + (\Delta y)^{2}$ .

Worked example. Distance from $A(2, 3)$ to $B(7, 15)$ .

$\Delta x = 5$ , $\Delta y = 12$ .
$d = \sqrt{25 + 144} = \sqrt{169} = 13$ .

3D extension. For points in 3D:

$d = \sqrt{(\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2}}$ .

Worked qualitative. What's the distance from origin to $(3, 4)$ ?

$d = \sqrt{9 + 16} = 5$ .
Directly $5$ , since $0$ -to-point distance.

Edexcel tip. Always show the differences explicitly. $\Delta x = 7 - 2 = 5$ . Mark scheme credits this step.

Pythagoras with $\Delta x, \Delta y$ .
$d = \sqrt{(\Delta x)^{2} + (\Delta y)^{2}}$ .
Extends to 3D.
Always compute differences.

How it’s examined

Pythagoras every Higher Tier paper (3-7 marks). Often combined with circles, cones, or coordinate geometry. Examiner reports flag (1) using on non-right triangles, (2) wrong direction (adding when should subtract), (3) forgetting square root.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H Jan 2024, 4MA1/2H May/Jun 2024. Last reviewed 2026-05-07.

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Step-by-step worked examples — Pythagoras Theorem

Step-by-step solutions to past-paper-style questions on pythagoras theorem, written exactly the way a tutor would explain them at the board.

1Find the hypotenuse
Foundation• Pythagoras, hypotenuse
Question
Right triangle: legs $3$ and $4$ . Find the hypotenuse.
Step-by-step solution
1. Step 1
  $a^{2} + b^{2} = c^{2}$ .
2. Step 2
  $3^{2} + 4^{2} = 9 + 16 = 25$ . So $c = 5$ .
Answer
$c = 5$
2Find a missing leg
Foundation• Adapted from 4MA1/1H Jan 2024 Q9• Pythagoras, leg
Question
Hypotenuse $13$ , one leg $5$ . Find the other leg.
Step-by-step solution
1. Step 1
  $5^{2} + b^{2} = 13^{2}$ .
2. Step 2
  $25 + b^{2} = 169 \implies b^{2} = 144 \implies b = 12$ .
Answer
$b = 12$
Examiner tip
Subtract from hypotenuse² when finding a leg. $b^{2} = c^{2} - a^{2}$ .
3Distance between two points
Higher• distance
Question
Find the distance between $A(2, 3)$ and $B(7, 15)$ .
Step-by-step solution
1. Step 1
  Form a right triangle: horizontal leg $\Delta x = 5$ , vertical leg $\Delta y = 12$ .
2. Step 2
  Pythagoras: $d^{2} = 5^{2} + 12^{2} = 25 + 144 = 169$ .
3. Step 3
  $d = 13$ .
Answer
$d = 13$
43D Pythagoras
Higher• Adapted from 4MA1/2H May/Jun 2024 Q18• 3D, Pythagoras
Question
Cuboid $4 \times 6 \times 12$ . Find the long diagonal.
Step-by-step solution
1. Step 1
  Floor diagonal: $4^{2} + 6^{2} = 52$ , so floor diag $= \sqrt{52}$ .
2. Step 2
  Long diag: $52 + 12^{2} = 52 + 144 = 196$ . So $d = 14$ .
3. Step 3
  Or directly: $d^{2} = 4^{2} + 6^{2} + 12^{2} = 16 + 36 + 144 = 196$ . $d = 14$ .
Answer
$d = 14$
Examiner tip
3D Pythagoras: $d^{2} = a^{2} + b^{2} + c^{2}$ . Same as 2D, just three terms.
5Pythagoras in an isosceles triangle
Higher• Pythagoras, isosceles
Question
Isosceles triangle with two equal sides $13$ cm and base $10$ cm. Find the perpendicular height.
Step-by-step solution
1. Step 1
  Drop perpendicular from apex to base. Bisects base: half = $5$ cm.
2. Step 2
  Right triangle with hypotenuse $13$ , leg $5$ :
  $h^{2} = 13^{2} - 5^{2} = 169 - 25 = 144$
3. Step 3
  $h = 12$ cm.
Answer
$h = 12$ cm

Key Formulae — Pythagoras Theorem

The formulae you need to memorise for pythagoras theorem on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

Pythagoras' theorem
$a^{2} + b^{2} = c^{2}$
$a, b$
two shorter sides (legs)
$c$
hypotenuse (opposite right angle)
When to use
Right-angled triangles.
Example
$3^{2} + 4^{2} = 25 = 5^{2}$ .
3D Pythagoras
$d^{2} = a^{2} + b^{2} + c^{2}$
$a, b, c$
three sides of cuboid
When to use
Long diagonal of a cuboid.
Example
$4 \times 6 \times 12$ cuboid: $d^{2} = 196$ , $d = 14$ .
Distance between two points
$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$
When to use
Coordinate geometry.
Example
$(2,3)$ to $(7, 15)$ : $\sqrt{25 + 144} = 13$ .

Key Definitions and Keywords — Pythagoras Theorem

Definitions to memorise and the exact keywords mark schemes credit for pythagoras theorem answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Hypotenuse
Examiner keyword
The longest side of a right-angled triangle. OPPOSITE the right angle.
Leg (of right triangle)
One of the two shorter sides forming the right angle.
Pythagorean triple
Three integers $a, b, c$ with $a^{2} + b^{2} = c^{2}$ . Common: $(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25)$ .

Common Mistakes and Misconceptions — Pythagoras Theorem

The traps other students keep falling into on pythagoras theorem questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Using $a^{2} + b^{2} = c^{2}$ when finding a leg
Why it happens
Memorising one direction.
How to avoid it
If finding the HYPOTENUSE: $c^{2} = a^{2} + b^{2}$ . If finding a LEG: $a^{2} = c^{2} - b^{2}$ .
✕Applying Pythagoras to non-right triangles
Why it happens
Forgetting condition.
How to avoid it
Pythagoras requires a RIGHT ANGLE. For other triangles, use cosine rule.
✕Forgetting to take square root at the end
Why it happens
Stopping after computing $c^{2}$ .
How to avoid it
Pythagoras gives $c^{2}$ . Square root to get $c$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Video lesson

Short walkthrough of the concepts students most often get stuck on.

Pythagoras Theorem — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Pythagoras Theorem

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Find the hypotenuse

2Find a missing leg

3Distance between two points

43D Pythagoras

5Pythagoras in an isosceles triangle

Pythagoras' theorem

3D Pythagoras

Distance between two points

Hypotenuse

Leg (of right triangle)

Pythagorean triple

✕Using a2+b2=c2a^{2} + b^{2} = c^{2}a2+b2=c2 when finding a leg

✕Applying Pythagoras to non-right triangles

✕Forgetting to take square root at the end

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Pythagoras Theorem — frequently asked questions

✕Using $a^{2} + b^{2} = c^{2}$ when finding a leg