Finding house width from a roof triangle and angle

Key concept: This is a right-triangle trigonometry question. The key is to identify the half-width and relate it to the roof angle and height.

Step by step

Identify the triangle in the roof diagram

The roof forms an isosceles triangle. If the vertical height is $3\text{ m}$ and each base angle is $26^\circ$, then splitting the roof down the middle gives two congruent right triangles.

In one half, the opposite side is $3\text{ m}$ and the adjacent side is half the house width. Let that half-width be $w$.

Then

$\tan 26^\circ = \frac{3}{w}.$

So

$w = \frac{3}{\tan 26^\circ}.$

Calculate the full width

The full width is twice the half-width:

$\text{width} = 2w = \frac{6}{\tan 26^\circ}.$

Using a calculator,

$\frac{6}{\tan 26^\circ} \approx 12.3.$

Rounded to the nearest metre, the house is

$\boxed{12\text{ m}}.$

Why this setup works

The angle is measured from the base, so tangent is the correct ratio to use. You do not need sine or cosine unless the diagram gives the sloping roof length instead of the height. The key step is recognizing that the roof is symmetric, which lets you use half the width rather than trying to work with the entire triangle at once.

Pitfall alert

A common mistake is to use the full width as the adjacent side in the tangent ratio. That gives an answer that is exactly twice too small. Another error is to measure the $26^\circ$ angle from the vertical instead of the horizontal base, which changes the trigonometric ratio you should use. Always check which side the angle touches in the diagram before writing the ratio.

Try different conditions

If the roof height were $4\text{ m}$ instead of $3\text{ m}$ while the angle stayed $26^\circ$, the same method would give width $=\frac{8}{\tan 26^\circ}\approx 16\text{ m}$. If the question gave the sloping side length instead of the height, then you would use cosine or sine rather than tangent. The triangle model stays the same; only the known side changes.

Further reading

right triangle trigonometry, tangent ratio, isosceles triangle