Pythagoras and isosceles triangles
Architects and engineers often use isosceles triangles when designing roofs, bridges, and supports. Pythagoras' Theorem helps calculate missing lengths—like the slant height of a roof—when only the base and height are known. Jump to the questions
Practise now
Calculate the missing lengths for these isosceles triangles. All answers should be rounded to one decimal place.
Topic guide
What this worksheet practises
This worksheet focuses on applying Pythagoras' Theorem to solve problems involving isosceles triangles. Pythagoras' Theorem (a² + b² = c²) only works on right-angled triangles. An isosceles triangle does not have a right angle, so you must create one first.
Key method
The secret is to slice the isosceles triangle in half.
- Draw a straight vertical line from the top point (the apex) straight down to the middle of the base.
- This line represents the height of the triangle. It cuts the isosceles triangle into two perfectly identical right-angled triangles.
- Crucial Step: Halve the length of the original base. This gives you the base length for your new right-angled triangle.
- Use Pythagoras' Theorem on this new right-angled triangle to find either the height or the sloping side.
Worked example
An isosceles triangle has a base of 10cm and two equal sloping sides of 13cm. Find the height of the triangle.
Step 1: Cut the triangle in half. The base of our new right-angled triangle is half of 10.
New base = 5cm.
Step 2: Identify the sides of the right-angled triangle. We have the base (a=5) and the hypotenuse (c=13). We need to find the height (b).
Step 3: Set up Pythagoras.
5² + b² = 13²
25 + b² = 169
Step 4: Solve for b.
b² = 169 − 25
b² = 144
b = √144 = 12.
The height of the triangle is 12cm.
Common mistakes to avoid
The most devastating mistake is forgetting to halve the base before starting Pythagoras. If a student uses the full base of 10cm alongside the hypotenuse of 13cm, the calculation will be completely wrong. Always remember: Pythagoras only works on right angles.
How to check your answer
The height of an isosceles triangle must always be slightly shorter than the sloping sides. In our example, a height of 12cm is mathematically sensible compared to the sloping side of 13cm. If you calculated a height of 15cm, you would instantly know an error was made.