What You'll Master
- Pythagoras Theorem: Linking the three sides of right-angled triangles using a² + b² = c²
- SOHCAHTOA: Finding missing sides and angles using sin, cos, and tan ratios
- Angles of Elevation & Depression: Solving real-world problems using horizontal reference lines
- Exact Trig Values: Memorising sin/cos/tan for key angles (0°, 30°, 45°, 60°, 90°) to answer without a calculator
Pythagoras Theorem
What Is It?
Pythagoras' theorem is a relationship between the three sides of any right-angled triangle. It tells you that the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides.
The hypotenuse is always the side opposite the right angle—it's the longest side, and you can spot it by looking for the small square symbol (□) which marks the 90° angle.
Finding the Hypotenuse
If you know the two shorter sides, finding the hypotenuse is straightforward:
c = √(a² + b²)
The steps:
- Square both shorter sides
- Add them together
- Take the positive square root
Finding a Shorter Side
If you know the hypotenuse and one shorter side, you need to rearrange the formula to make the unknown the subject.
a = √(c² − b²)
The steps:
- Square the hypotenuse and the known shorter side
- Subtract (bigger value minus smaller value—this is crucial!)
- Take the positive square root
⚠️ Critical Mistake: When finding a shorter side, you subtract inside the square root, not add. If you get a negative number before the square root, you've made an error—check which number is larger.
Worked Example 1: Finding the Hypotenuse
A right-angled triangle has sides of 5 cm and 12 cm. Find the hypotenuse.
Step 1: Identify what you have
a = 5 cm, b = 12 cm, c = ?
Step 2: Use the formula c = √(a² + b²)
c = √(5² + 12²)
Step 3: Calculate inside the square root
c = √(25 + 144) = √169
Step 4: Find the square root
c = 13 cm
Answer: 13 cm
Try This (Finding the Hypotenuse)
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the length of the hypotenuse.
Worked Example 2: Finding a Shorter Side
A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the other shorter side.
Step 1: Identify what you have
c = 13 cm, a = 5 cm, b = ?
Step 2: Use the formula a = √(c² − b²)
b = √(13² − 5²)
Step 3: Calculate inside the square root (subtract, don't add!)
b = √(169 − 25) = √144
Step 4: Find the square root
b = 12 cm
Answer: 12 cm
Try This (Finding a Shorter Side)
A right-angled triangle has a hypotenuse of 25 cm and one shorter side of 7 cm. Find the length of the other shorter side.
Using Pythagoras with Other Shapes
Pythagoras' theorem only works with right angles, but you can use it on any shape that contains a right-angled triangle. For example:
- Rectangles: Draw a diagonal to split it into two right triangles
- Cuboids or pyramids: Identify internal right angles and apply the formula
Exam Tip:
leave your answer as an exact value
SOHCAHTOA & Trigonometry
What Is Trigonometry?
sine