Maths IA · Geometry and trigonometry · topic 27 of 40

Building height from angles: how accurate is it? · IB Maths IA idea: guide, sample graph and step-by-step plan

outdoor maths: good for engagement

Anyone can write H = d tan θ. The real question is how far you can trust the answer, and which measurement lets you down first.

Difficulty●○○ Gentle
KitClinometer (phone app), tape measure
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like going outside to measure
  • ✓ you can handle trigonometry
  • ✓ you want a clear way to show 'how good is this?'

The big idea

← the error work is where marks live

Measuring a tall building from the ground is trigonometry: an angle of elevation and a distance, or a shadow and the Sun's angle. Pythagoras and a right-angled triangle are prior learning, so the exploration lives in the error work. How does a small error in the angle grow into an error in the height? Where is the method at its best and worst? Compare two methods, predict your uncertainty before measuring, then test it against a height you can check. Strong work derives how the error depends on the angle, not only on the data.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Derive the error formula

Use differentiation to find how an error in θ and in d feeds into the height.

→ A formula for the uncertainty that depends on the angle: you can predict the best angle to measure from before you leave home.
Compare two methods

Angle of elevation against shadow length with the Sun's angle (or a vertical stick of known height).

→ Two sets of errors to compare, with a judgement about which method is better, and when.
Take readings from many distances

Measure the same building from five or six distances.

→ You can graph the spread in height against distance or angle, and test the error model against it.
Use a building you can check

A building with a published height, or one whose height you can measure with a long tape or from floor plans.

→ A true value to compare against, so you can say if the error bars were honest.
Handle sloping ground

Measure from ground that is not level.

→ Extra geometry: you need two angles and a baseline, or the sine rule, and extra error sources.
Simulate the errors

Random errors on each measurement, many times, using a spreadsheet or code.

→ Compare your formula for the error with a simulation, and discuss where a first-order formula starts to fail.

Your Maths IA research question: run the check

what does 'accurate' mean? say it

Write your own question first. Then make sure it ticks every box:

  • Names the object and the method (or methods) you will compare.
  • Says what you will measure and with which instrument, including its resolution.
  • Says what 'accurate' means here: an uncertainty you calculate, a comparison with a known value, or both.
  • Names the thing you will vary (distance? angle? method?) so there is something to analyse.
  • Is small enough for 12 to 20 pages and lets the error maths do the heavy lifting.
  • Can be answered with numbers and units, not with a general statement.

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The picture to draw

A right-angled triangle with the distances you can measure and the one you cannot. Draw your own with your real numbers, and add the shadow method if you use it.

θdsH = ?clinometer reading θeye height hsun ray:second methodflat ground, tape measure

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Title page, question and rationale~1 page
Your question; which building you chose and why it matters to you.
Aim and plan~0.5 page
What you will measure, how, and how you will judge 'how accurate'.
The model~1.5 pages
The height formula for each method, with a clear diagram and defined symbols. Include the eye-height correction.
Measurements~2 pages
Your method, instruments and resolutions, and a table of raw readings with units.
Error work~3 pages
A derivation of how errors in θ, d (and h) feed into H, from differentiation or a worst-case argument; a prediction before you look at your data. At HL this is where the HL-level maths must show.
Results and comparison~3 pages
Heights from each method with their uncertainties; a graph of uncertainty against angle or distance; comparison with a known height if you have one.
Testing the error model~2 pages
Do your data match the predicted uncertainty? A simulation or a statistical comparison.
Reflection throughout~1.5 pages
Which assumptions failed (level ground, vertical wall, straight sight line), and what each would do to the answer.
Conclusion~0.5 page
Answer with numbers, and say which method you would use and when.
Bibliography and appendixnot counted
Sources, extra tables and readings.

What your Maths IA graph could look like

curve from maths, points from you

Made-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only

1020304050607080024681012Angle of elevation θ / ° (± 1°)Uncertainty in H from θ alone / %your derived uncertainty curve (δθ = 1°)spread of your repeats, with error barswhere is the anglemost forgiving?why?ILLUSTRATIVE · MADE-UP DATA
  • The curve comes from a derivation, then the data tests it: the same story the maths should tell.
  • Axis labels say the quantity, its unit and the uncertainty of the variable you set.
  • The scale starts at 0 so the reader sees how large the best case still is.
  • Both axes carry uncertainty: vertical bars for the spread, horizontal bars for the angle resolution.
  • Another graph to consider: uncertainty in H from d against distance, or both sources added in quadrature.

Which measurement hurts most?

rank them: don't just list them

Rank the sources by what they do to the result H, not by how big they look. A chart like this, with your numbers, shows what to measure better first.

Angle θ (± 1°)3.3%Distance d (± 0.2 m)0.63%Eye height h (± 0.05 m)0.19%biggest effect: discuss this first!% uncertainty in H: d = 30 m, θ = 40°ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 11 done

  1. Read and predictWrite down the formula for each method. Before you measure, predict which one will be better and by roughly how much.
  2. Derive the error behaviourDifferentiate H with respect to each measured quantity. Work out where the angle error matters least and most. This is the heart of the maths.
  3. Trial run (one afternoon)Measure a small building or tree you can check. Is your angle reading steady? Can you measure d on flat ground? Fix the method now.
  4. Choose your building and positionsPick a target and a range of distances or angles, with reasons for each. Note the ground and the light.
  5. Collect the dataRecord raw angles and distances with units and instrument resolution. Take repeats at each position.
  6. Calculate heights and uncertaintiesOne worked example, then a table. Check the units and that rounding does not creep in.
  7. Graph and compareGraph the uncertainty against angle or distance. Does the pattern follow your derived curve? Compare the two methods by a number.
  8. Test the modelUse a known height, a simulation, or both. Where does a first-order error formula begin to fail?
  9. Reflect as you goWhich assumption worries you most, and what would it do to the answer?
  10. Write, then cut12 to 20 pages. Show each calculation type once. Axis titles, units and notation (no computer syntax) all need checking.
  11. Check before your teacher's draftDoes your conclusion say whether 'how accurate' was answered, with numbers?

Maths you'll need

Traps to avoid in this Maths IA

  • Stopping at H = d tan θ. That is prior learning, and it is the whole 'method' with no analysis.
  • Measuring once from one place and calling the answer accurate.
  • Only quoting 'human error'. Say which measurement and how big, as a number.
  • At HL, using SL-level trigonometry only: add differentiation, partial changes or a simulation of the errors.
  • Forgetting the height of the eye (or the instrument) above the ground.
  • Assuming the ground is level and the wall vertical without checking.
  • Quoting the height to the nearest millimetre when the angle only allows the nearest metre.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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