Lesson 4 of 6 · Degrees, radians & right-triangle geometry
Turn a viewing angle into a length
8 minute lesson · 3 checks along the way
Your goal for this lessonIdentify the sides of a right triangle, choose a trigonometric ratio, and distinguish horizontal range from slant range.
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State the angle and its reference
An angle describes a change of direction, so its reference direction matters. An elevation of 30 degrees above a local horizontal line differs from an angle of 30 degrees away from the vertical. A full turn is 360 degrees or 2π ; 180 degrees equals π radians. Both units describe the same kind of quantity. Check a calculator’s mode before entering a numerical angle.
A radian relates arc length to the radius of a circle. This makes it especially useful in equations involving circles and angular rates, but you can begin triangle work in degrees. Here all numerical angles are in degrees unless stated otherwise. The symbol θ, pronounced theta, simply stands for the chosen angle. It is a label for a quantity, not a new measurement technique.
Choose the sides relative to your angle
In a right triangle, the lies opposite the right angle and is the longest side. For the acute angle θ you selected, name the other sides opposite and adjacent. The ratios are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. Rotate the sketch and those relationships remain valid. Switch to the other acute angle and the opposite and adjacent labels exchange.
Consider an original survey exercise on level ground. A camera is 20 meters horizontally from a vertical marker and measures its top at 30 degrees above the camera’s horizontal line. We want the height above that line, so the relevant sides are opposite and adjacent. Use h = 20 m × tan 30°. With tan 30° ≈ 0.577, the top is about 11.5 meters above the camera.
Label the reference level
Original idealized survey diagram, drawn from the lesson’s supplied quantities. The camera height is added only when the question asks for height above ground.
Open full-size diagram in a new tab: Label the reference levelCheck 1 of 3
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Work through a decision
Solve the right height problem
If the camera sits 1.5 meters above the same level ground, add that height: the marker top is about 13.0 meters above ground. Forgetting the camera height answers a different question. The diagram labels the reference level and the right angle so you can check the calculation. These numbers describe a simplified training target, not a surveyed feature in the Spirit photograph.
Now change the supplied measurement: a laser reports a 25-meter slant distance to a target at 30 degrees elevation. The slant distance is the hypotenuse, so its vertical is 25 m × sin 30° = 12.5 meters. Using with that 25-meter value would mistakenly treat the sloping line as horizontal. Name each measured length before selecting the formula, even when the numerical angle looks familiar.
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Know what an image does not supply
Spirit’s hazard-camera view shows wheels, lander structure, and terrain through a wide-angle optical system. It is an observation, not a flat engineering drawing. Measuring a line’s screen angle with a protractor does not automatically recover its angle in three-dimensional space. Camera calibration, viewpoint, distortion, and the geometry of the scene must connect image coordinates with physical directions before a survey calculation is justified.
Our triangle assumes level ground, a vertical target, a known horizontal baseline, and negligible curvature over the small scene. A distant hill, a tilted camera, or uncertain range may require a different model and an uncertainty estimate. Trigonometry helps turn appropriate measurements into missing lengths; it cannot create the measurements. Record the angle convention and reference level with the result so another reader can reconstruct the same triangle.
The archive preserves a still from NASA’s PIA05040 hazard-camera sequence documenting Spirit’s wheel-base deployment. NASA’s original release alternates before and after views; this saved still does not show the full sequence. The wide-angle scene is not the flat survey triangle used in the exercise.
Look closely
Can a protractor applied to this displayed image recover a terrain elevation angle without camera information?
See what to notice
No. Image coordinates depend on the camera’s optical mapping and orientation as well as the scene. The triangle exercise instead provides a horizontal baseline, an elevation angle, and a level-ground reference explicitly.
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Put it together
Your lesson progress
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