What this guide covers
This guide covers a missing side in a right triangle, including triangles embedded in a larger diagram. It does not cover the whole trigonometry topic or assume that a sketch proves a right angle.
Add or subtract? Write the relation first
If c is the hypotenuse and a and b are the two shorter sides, a² + b² = c².
- Missing hypotenuse: c = √(a² + b²). Add the squared shorter sides.
- Missing shorter side: a = √(c² − b²). Subtract from the squared hypotenuse.
- Rotating the diagram changes neither the right angle nor which side is opposite it. The hypotenuse is not always the sloping side on the page.
Find the hypotenuse
A right triangle has perpendicular sides of 8 cm and 15 cm. Find its hypotenuse h.
- The missing side is opposite the right angle, so h² = 8² + 15².
- h² = 64 + 225 = 289.
- h = 17 cm; use the positive root because h is a length.
Check: 17 is longer than either leg, and 8² + 15² = 17².
Find a shorter side
A right triangle has hypotenuse 25 cm and one shorter side 7 cm. Find the other side x.
- Keep the hypotenuse on its own: x² + 7² = 25².
- x² = 625 − 49 = 576.
- x = 24 cm.
Check: 24 is less than 25. Adding 625 and 49 instead would calculate the wrong side role.
When the triangle is inside another shape
School-paper examples combine Pythagoras with a preceding length calculation or a right-triangle section of a solid. Separate finding the triangle from calculating its side.
- Identify the three sides of the particular triangle you are using.
- Use a marked right angle or a justified perpendicular relationship; do not rely on appearance.
- If a length must be found first, label it before substituting.
- For a cone cross-section, distinguish radius, perpendicular height and slant height. A diameter is not the radius.
Check the method, then the number
A correct numerical answer can conceal a guessed setup. Keep the squared relation in your working.
- Use a² + b² = c², not a + b = c.
- Take the square root after adding or subtracting the squares.
- The hypotenuse must be the longest side. A shorter-side answer larger than it signals a problem.
- Use common units before calculating and avoid rounding an intermediate length too early.
Try it yourself
Work on paper, then compare the steps. These checks do not establish mastery.
The right angle is at B in triangle ABC. Which side is the hypotenuse?
AC, because it is opposite B. The letters and orientation do not change the rule.
The perpendicular sides are 9 cm and 12 cm. Find the hypotenuse.
√(9² + 12²) = √225 = 15 cm.
A right triangle has hypotenuse 17 cm and a leg of 15 cm. Find the remaining leg.
√(17² − 15²) = √64 = 8 cm. Start from x² + 15² = 17².
Where this connects
- Explore the Sec 2 curriculum topics →
See where Pythagoras sits beside the other curriculum topics.
- Explore the connected maths map →
See the wider connections without treating every connection as a mastery gate.
- Later connection: distance between coordinates →
Horizontal and vertical displacements form the legs of a right triangle. The existing guide covers wider Upper Secondary work too.
Sources and scope
This focused Sec 2 guide is informed by school-paper observations. Reviewed tasks include Ngee Ann Sec 2 2025 Q17(a) and Canberra Sec 2 2022 Paper 2 Q12(a). These are two observed school contexts, not a universal teaching timetable or a national frequency estimate. Finding the hypotenuse and finding a shorter side use the same relation. Examples here are original. A personalised Sec 2 diagnostic is not available.