Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #4
Grade 3 geometry-2d
Pick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The outer triangle is built from two kinds of pieces: one small triangle plus three trapezoids, with nothing overlapping. Tool #7 (Identify Subproblems) breaks the goal into two easy steps — first find the total area taken by all three trapezoids (subtract the inner triangle from the outer), then split that area evenly among the three congruent trapezoids. Tool #15 (Visualize) confirms the picture: the three trapezoids tile the ring-shaped region between the triangles with no gaps and no overlap, so their areas truly add up to the difference.
Picture the two triangles
The outer triangle is the inner triangle plus three non-overlapping trapezoids, so its area is the sum of those parts.
Grade 3 area work: when a region is split into non-overlapping pieces, the whole area equals the sum of the parts.
3.MD.C.7Organize Information In More WaysSubtract the inner triangle
Subtract the inner triangle from the outer triangle to get the three trapezoids' combined area, 15.
This is the "area of the ring" subproblem: the trapezoids fill exactly the space the small triangle doesn't.
3.MD.C.7Identify SubproblemsSplit the area three ways
The three trapezoids are congruent, so split that combined area equally by dividing it by 3.
Grade 3 partitive division: 15 shared equally among 3 identical pieces gives 5 each.
The area of one trapezoid equals one third of the difference between the outer triangle's area and the inner triangle's area.
▸ Why?
The three trapezoids together fill exactly the region between the two triangles, and that between-region's area is the outer area minus the inner area.
▸ Why?
The outer triangle is made up of the inner triangle plus the three trapezoids, with no gaps and no overlaps, so its area is the sum of those parts.
▸ Why?
Removing the inner triangle's area from the outer area undoes having added it, which leaves the three trapezoids' combined area.
▸ Why?
The three trapezoids are congruent, so their combined area splits into three equal shares and each trapezoid is one of those shares.
▸ Why?
Each trapezoid can be slid, turned, or flipped to lie exactly on another, so the three of them cover the same amount of area.
▸ Why?
The combined area is three equal groups of one trapezoid, so one trapezoid is that combined area divided by three.
▸ Why?
Three equal trapezoids form three equal groups, so their total is three times the area of one trapezoid.
▸ Why?
Dividing the total by three undoes multiplying one trapezoid by three, recovering the area of a single trapezoid.
Big triangle minus small triangle gives the trapezoids' combined area — then divide by 3 because they are identical. Subtract, then share equally.
- Picture the two triangles
- Subtract the inner triangle
- Split the area three ways
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2024 #3 Gr 3Similar level
Same sub-type: Nested Frame / Concentric DifferenceSame techniqueArea DifferenceFour squares of side length 4, 7, 9, and 10 are arranged in increasing size order so that their left edges and bottom ed…
- AMC 8 2000 #6 Gr 3Similar level
Same techniqueArea Difference Identify SubproblemsFigure ABCD is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area o…
- AMC 8 2005 #13 Gr 3Similar level
Same techniqueArea Difference Identify SubproblemsThe area of polygon ABCDEF is 52 with AB=8, BC=9 and FA=5. What is DE+EF?