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Broken tree height: why the sloping part is not the whole answer

5 min read

Question

A tree breaks so that its top touches the ground 8 m from the foot of the tree. The broken part makes an angle of 30° with the ground. Find the original height of the tree.

Broken tree height? Don't forget the bent part!Watch on YouTube

The answer to this broken-tree problem is 83 m8\sqrt3\text{ m}. The trigonometry is straightforward. The trap is deciding what the question means by the tree's original height.

After the break, the tree has two pieces: the vertical stump that is still standing and the sloping piece that fell. Finding either piece alone cannot answer the question.

30°8 mstump = sfallen part = foriginal heights + foriginal top
The dashed vertical line reconstructs the original tree. Its height is the standing stump plus the fallen part—not either length on its own.

Step 1: find the standing stump

In the right triangle, the 8 m ground distance is adjacent to 30°, while the stump is opposite it. That makes tangent the cleanest ratio:

tan⁡30∘=oppositeadjacent\tan 30^\circ=\frac{\text{opposite}}{\text{adjacent}}
tan⁡30∘=stump8\tan 30^\circ=\frac{\text{stump}}{8}
13=stump8\frac{1}{\sqrt3}=\frac{\text{stump}}{8}
stump=83 m\text{stump}=\frac{8}{\sqrt3}\text{ m}
stump=83×33 m\text{stump}=\frac{8}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}\text{ m}
stump=833 m\text{stump}=\frac{8\sqrt3}{3}\text{ m}
stump≈4.62 m\text{stump}\approx4.62\text{ m}

Step 2: find the fallen piece

The fallen piece is the hypotenuse of the same right triangle. The 8 m side is still adjacent to 30°, so use cosine:

cos⁡30∘=adjacenthypotenuse\cos 30^\circ=\frac{\text{adjacent}}{\text{hypotenuse}}
cos⁡30∘=8fallen piece\cos 30^\circ=\frac{8}{\text{fallen piece}}
32=8fallen piece\frac{\sqrt3}{2}=\frac{8}{\text{fallen piece}}
fallen piece=83/2 m\text{fallen piece}=\frac{8}{\sqrt3/2}\text{ m}
fallen piece=163 m\text{fallen piece}=\frac{16}{\sqrt3}\text{ m}
fallen piece=163×33 m\text{fallen piece}=\frac{16}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}\text{ m}
fallen piece=1633 m\text{fallen piece}=\frac{16\sqrt3}{3}\text{ m}
fallen piece≈9.24 m\text{fallen piece}\approx9.24\text{ m}

Step 3: rebuild the original height

original height=stump+fallen piece\text{original height}=\text{stump}+\text{fallen piece}
original height=833+1633\text{original height}=\frac{8\sqrt3}{3}+\frac{16\sqrt3}{3}
original height=2433\text{original height}=\frac{24\sqrt3}{3}
original height=83 m\text{original height}=8\sqrt3\text{ m}
original height≈13.86 m\text{original height}\approx13.86\text{ m}

Numerically, 83≈13.868\sqrt3\approx13.86, so the original tree was about 13.86 m tall.

A faster route once the diagram is understood

Because the angle is 30°, the side opposite 30° is half the hypotenuse. Here, the stump is opposite 30° and the fallen piece is the hypotenuse. Therefore the fallen piece is twice the stump. The original height is three stump lengths:

fallen piece=2×stump\text{fallen piece}=2\times\text{stump}
original height=stump+2×stump\text{original height}=\text{stump}+2\times\text{stump}
original height=3×stump\text{original height}=3\times\text{stump}
original height=3(833)\text{original height}=3\left(\frac{8\sqrt3}{3}\right)
original height=83 m\text{original height}=8\sqrt3\text{ m}

This shortcut is safe only after you have correctly identified the sides. Memorising “multiply by three” without the 30° geometry will fail as soon as the angle changes.

Four exam checks

  • Use the angle with the ground. The angle with the vertical stump would be 60°.
  • Do not call 8 m the hypotenuse. It is the horizontal, adjacent side.
  • Keep surds exact. Round only after reaching the final expression.
  • Check the scale. If the 8 m distance doubled at the same angle, every length—and the original height—would double.

Young Math Brains makes you build the right triangle and choose the ratio before any formula appears, then gives step-by-step practice for single-triangle height and distance problems.

Practise heights and distances interactively

Your turn

Before calculating a broken-tree question, which two physical lengths must be added to reconstruct the original height?

Tell us in the comments on YouTube or Instagram, or email connect@youngmathbrains.com. We read every one.

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