(In all solutions below, the tree or other landmark on the opposite side of the river is at point A, while the point directly across the river from it (on the near side) is called point B. Solutions almost always begin by having someone make a right angle at point B by walking up or downstream from B beside the river.)
- Solution 1: scale map
Walk along the river an arbitrary distance to point C. Measure length BC and angle ACB (angle ABC should measure 90 degrees). Then make a scale map on paper, duplicating the angle measurements. Now measure the desired length AB on the map, and use the scale (proportion) to calculate the length of the real AB.
Source: student answer site for United We Solve published by eeps - Solution 2: similar right triangles on the ground
Walk along the river an arbitrary distance to point C (and mark it). Then turn 90 degrees and walk away from the river an arbitrary distance, to point D. Mark point E at the intersection of lines BC and AD (this will probably require people stationed at B, C and D, and a fourth person adjusting location until (s)he is on both lines. Now triangles ABE and DCE are similar. Measure lengths BE, EC, and CD, and find AB by proportion.
Source: Wayne State University's Problem of the Week 8, from fall 1999 - Solution 3: congruent right triangles on the ground
Very similar to Solution 2, but a little extra care makes the triangles congruent and avoids the need to calculate. Walk along the river an arbitrary distance to point E (and mark it). Now continue walking along line BE (from E) and duplicate length BE, to a new point C with BE=EC. Then turn 90 degrees and walk away from the river until you can see point E line up with point A. This new point D forms a triangle DCE which is congruent (not just similar) to ABE. Measure DC, which equals AB.
Source: Math Forum - Solution 4: congruent right triangles through a mirror
Stand immediately across the river (point B) from the tree (point A). Your friend walks some distance along the shore perpendicular to the line from you to the tree (i.e., perpendicular to line AB) and (from this new point C) holds up a mirror parallel to this line, facing you (standing at point B, you can see yourself in it. Wave hello to yourself!). You now walk away from the tree and river along line AB until you can see the tree (point A) reflected in the mirror (which is still at point C). At this point (point D) you are exactly as far from point B as the tree is! One variation of this solution used in class (without a mirror) was to have the person at point C measure angle ACB and then tell a person walking along line AB away from the river when to stop (at point D) so that angle BCD has the same measure as angle ACB. An interesting point is that this solution doesn't actually require the angle to be measured in units, merely copied.
Source: Kribs-Zaleta - Solution 5: special right triangles
There are two "special" right triangles known by geometry students, the 45-45-90 and 30-60-90 triangles (so called because those are the vertex angle measurements in degrees). The 45-45-90 triangle is isosceles, and you can recreate one by walking along the river from point B, perpendicular to line AB, until the angle between the lines from where you are to points A and B is exactly 45 degrees. Then you are at a point C which is exactly as far from point B as point A is!
The 30-60-90 triangle is half an equilateral triangle, and is known to have sides in proportion 1, square root of 3, and 2 (the hypotenuse). To recreate this, walk along the river from point B, perpendicular to line AB, until the angle between the lines from where you are to points A and B is exactly 60 degrees (this will involve less walking than getting to 45 degrees). Then you are at a point C which forms a 30-60-90 triangle ACB, with the desired length AB exactly the square root of 3 times length BC.
Source: Kribs-Zaleta - Solution 6: right triangle trigonometry (tangent)
Walk along the river an arbitrary distance to point C. Measure length BC and angle ACB (angle ABC should measure 90 degrees). Then write the equation tanACB=AB/BC, substitute, and solve for AB.
Source 1: learner.org's measurement course, Session 5C
Source 2: The Math Page
Source 3: a story from the Peace Corps in Togo! --- if you don't read any others, read this one! - Solution 7: Laws of Sines and Cosines
Pick two arbitrary points B and C on the near side, two more points D and E such that ABD and ACE are lines, measure all the lengths on the near side of the river, and blast away with the Laws of Sines and Cosines. Requires absolutely no angle measurements, but a lot of trig. The solution (see link) also discusses precision.
Source: River Width assessment from the Illinois Learning Standards performance assessments for Grades 11 and 12 mathematics - a related problem: Napoleon's River
the classic solution, and why it's impractical, from UC Davis
- Interactive activity from learner.org on determining a circle's area by decomposing and reassembling it
- learner.org's approach to Changing Linear Dimensions (includes video)
- learner.org discusses precision and significant figures
- Video on measuring areas of rectangles
- summary of Outhred & Mitchelmore study on children's array drawing/structuring
- Measuring Paragraphs
This link shows the same text set in columns of many different widths; you can measure the total area of text and see it is more or less conserved.
- The area of Texas is 268,601 square miles, or 620,307 triangular miles, or 691,200 sq.km., or 1,596,258 tr.km. (Actually, reports of Texas's area on the Internet vary, say between 678,000 and 696,000 sq.km.; the figures given here in triangular units are obtained simply by converting from the corresponding square units.)
- learner.org's approach to volumes of prisms and cylinders, pyramids and cones, includes an interactive activity comparing spheres and cones to cylinders
- Here's one approach to showing how 3 pyramids fit in a cube (see also the animation at right), but you can google 3 pyramids cube and you'll get half a million hits.
- boundary vs. interior posters
- Video on optimization, surface area and volume
- Estimating Your Hand meets Perimeter vs. Area at learner.org (Problem A7 at the top of the page)
- Calendars through the ages
- a photo of an old clock face showing why the minute hand is longer than the hour hand (and what it used to point to)
