Find the hidden triangle
Last time you calculated triangles someone had already drawn. But life does not draw triangles. Life gives you a ladder, a wall and the question “will it reach?”.
The real skill is not plugging into a formula. It is seeing a right-angled triangle where nobody drew one.
A ladder leaning on a wall: the wall is vertical, the ground is horizontal, there is a right angle between them. The ladder is the hypotenuse. A screen diagonal: the screen is a rectangle, the diagonal cuts it into two right-angled triangles. A shortcut diagonally across a park instead of walking two sides: again a hypotenuse.
The method is always the same: draw a sketch, find the right angle, label the sides, substitute, calculate, check whether the result makes sense.
Today we will walk through three such situations step by step. Then you will see triangles everywhere — and that is exactly the goal.
Sofie is buying a phone from a second-hand seller. The seller writes: “screen 16 cm tall, 7.5 cm wide”. But phones are sold by the diagonal and Sofie wants to know if it is the famous “7-inch” model. She draws a rectangle, a diagonal, and in her head she sees a triangle. 16² + 7.5² = 256 + 56.25 = 312.25. The square root is 17.7 cm. She divides by 2.54 (an inch is 2.54 cm): almost exactly 7 inches. “It fits. And the seller does not even know I recalculated it with Pythagoras.”
Always start with a sketch, even if it is ugly. Anyone who draws can see where the right angle is and which side is the hypotenuse. Anyone who does not draw is guessing.
Method for word problems
1. Draw the situation. Wall, ground, ladder. Three lines are enough.
2. Find the right angle. Vertical and horizontal things almost always make one: a wall and a floor, a pole and the ground, the sides of a rectangle.
3. Label the sides with numbers from the question. What is opposite the right angle is the hypotenuse. A ladder, a diagonal, a shortcut, a tight rope — slanted things are usually the hypotenuse.
4. Substitute into a² + b² = c² and calculate as last time.
5. Check the sense. The hypotenuse must come out longest. Does the ladder reach higher than its length? Mistake. A diagonal shorter than a side? Mistake.
When the numbers do not come out nicely (and in practice they almost never do), estimate the square root between neighbours and round — say to one decimal place. With measuring, nobody will notice a millimetre anyway.
Write units at every step. Metres must not meet centimetres in one formula — convert everything first.
Example 1: a ladder against a wall
The ladder is 5 m long and stands 1.4 m from the wall. How high does it reach?
Sketch: wall vertical, ground horizontal, ladder slanted — hypotenuse c = 5, shorter side a = 1.4, I am looking for shorter side b (the height).
b² = c² − a² = 25 − 1.96 = 23.04.
b = √23.04 = 4.8 m.
Sense check: 4.8 m is less than the ladder’s length 5 m. It fits — a slanted ladder always reaches lower than it is long.
Example 2: a shortcut across a park
The park is a rectangle 300 m × 400 m. Ema walks along two sides, Adam goes diagonally on a path. How much shorter is Adam’s walk?
The path is the hypotenuse: c² = 300² + 400² = 90,000 + 160,000 = 250,000, so c = √250,000 = 500 m.
Ema walks 300 + 400 = 700 m. Adam 500 m.
Difference: 700 − 500 = 200 m. The shortcut saves almost a third of the walk — that is why parks have worn paths going diagonally.
Example 3: a monitor diagonal
A monitor is 53 cm wide and 30 cm tall. It is sold as “24-inch”. Does that fit?
Diagonal: c² = 53² + 30² = 2809 + 900 = 3709.
c = √3709 ≈ ? Neighbours: 60² = 3600, 61² = 3721. So almost exactly 61 cm — more precisely 60.9.
Inches: 60.9 : 2.54 ≈ 24 inches. It fits. Shops give the diagonal because it is the biggest number they can write about a screen.
Do not mix units. If the wall is in metres and the distance is in centimetres, convert everything to one — otherwise you get nonsense that even a sense check will not save.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. البداية with ten. When it goes well, add more.
Find the missing side of a right-angled triangle. Picture a ladder or a diagonal with every question.
On paper
Three situations, three sketches. No sketch, no calculating.
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A ladder 6.5 m long stands 2.5 m from the wall. Draw a sketch and calculate how high it reaches.
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A rectangular pitch measures 60 m by 80 m. Calculate the diagonal and how many metres shorter it is than walking two sides.
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A kite flies on a tight string 50 m long. Filip stands 30 m from the spot directly under the kite. How high is the kite flying?
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Measure (with a ruler) the width and height of your phone screen, calculate the diagonal and compare with the maker’s figure in inches (1 inch = 2.54 cm).
Now you 💪
- I can draw a sketch with a right angle from the text of a problem.
- I recognise that a slanted thing (ladder, diagonal, rope) is usually the hypotenuse.
- After calculating I check whether the result makes sense.
Done when: You have three sketches with calculated sides, your own phone measured, and a run of five correct in practice.
What to take from this lesson
- Right angles hide between vertical and horizontal: a wall and the ground, the sides of a rectangle.
- Slanted things — a ladder, a diagonal, a shortcut — are usually the hypotenuse.
- Method: sketch → right angle → label the sides → substitute → check the sense.
- Unify the units first, then calculate.
- An ugly square root: estimate and round — for measuring that is enough.
© 2026 Ing. Martin Polak / AlgoRhino · شروط استخدام المحتوى