Find a practical angle
Use the requested reference line and two correctly interpreted lengths
- Ground angle For rise r and horizontal run d, use inverse tangent \(\displaystyle θ=\tan^{-1}(r/d)\)
- Measured sloping length A sloping length is the hypotenuse only when it lies opposite the established right angle. Use sine or cosine with the stated leg
- Raised reference Subtract heights measured from the same level ground before forming the rise-to-run quotient \(\displaystyle θ=\tan^{-1}\left(\frac{H-h_0}{d}\right)\)
Worked example
A ramp rises 1.8 m over 7.5 m horizontally. Find its angle above horizontal to one decimal place
\[\tan θ=\frac{1.8}{7.5}\]
- \(\displaystyle θ=\tan^{-1}\left(\frac{1.8}{7.5}\right)\)Use the original quotient in degree mode
- \(\displaystyle θ=13.495733\ldots^{\circ}\)Retain the unrounded calculator angle
- \(\displaystyle θ\approx13.5^{\circ}\)Round only the final angle
Common mistakes
Using the horizontal run as hypotenuse
✕\(\displaystyle θ=\sin^{-1}(1.8/7.5)\)✓\(\displaystyle θ=\tan^{-1}(1.8/7.5)\)
The 7.5 m is horizontal run, not ramp lengthIgnoring instrument height
✕\(\displaystyle θ=\tan^{-1}(H/d)\)✓\(\displaystyle θ=\tan^{-1}((H-h_0)/d)\)
For a raised instrument on level ground, the triangle rise is the height differenceDifficulty levels
standardFind the angle of a practical triangle from lengths in the same units
challengeFind a practical angle, including mixed-unit measurements
Questions about this topic
Which angle does the calculation find?+
The angle used to name opposite and adjacent. An angle measured from vertical is the complement of the angle from horizontal
Can I use rise divided by ramp length?+
Yes, with inverse sine when the ramp is the hypotenuse. Rise divided by horizontal run requires inverse tangent
Must the lengths use the same units?+
Yes. Convert first so the quotient is dimensionless
What assumptions does this model make?+
It uses straight segments and an established right angle. Uneven ground or a nonvertical support may require a different model
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 9: Pythagoras and trigonometry
In this chapter
9A Finding the hypotenuse: Find a side of a right-angled triangleDetermine whether three sides form a right-angled triangle9B Finding a shorter side: Find a side of a right-angled triangle9C Pythagoras' theorem in 2D and 3D: Find a side of a right-angled trianglePythagoras in three dimensionsName sides and state trigonometric ratios9D Trigonometric ratios: Evaluate a trigonometric ratio9D Trigonometric ratios: Choose a trigonometric ratio9E Unknown side in the numerator: Find an unknown side of a right-angled triangle9E Unknown side in the numerator: Solve a practical right-angled-triangle problem9E Unknown side in the numerator: Evaluate a trigonometric ratio9E Unknown side in the numerator: Choose a trigonometric ratio9F Unknown side in the denominator: Find an unknown side of a right-angled triangle9F Unknown side in the denominator: Solve a practical right-angled-triangle problem9F Unknown side in the denominator: Choose a trigonometric ratioFind an angle of a right-angled triangle from two sides9G Finding angles with trigonometry: Solve a practical right-angled-triangle problem9G Finding angles with trigonometry: Evaluate a trigonometric ratio, and find an angle from its valueAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsTrigonometry in three dimensions