Competition · AMC preparation · step 4 of 4
AMC 8 · 2025 · #1
Grade 6 geometry-2d
Pick an answer.
AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The star is a single complicated shape, but it is built from simple pieces that already line up with the grid: one central 2 × 2 square and eight half-square triangles. Tool #7 (Identify Subproblems) cleanly splits the star into "central square area" + "8 point-triangle areas", each of which is one elementary area calculation. Tool #1 (Draw a Diagram) helps mark which unit cells are fully shaded, which are half-shaded, and which are empty, so we do not double-count. After computing the star area we form the ratio star/total, convert to percent, and use Tool #3 (Eliminate Possibilities) to confirm the match with choice (B).
Find the board's total area
The 4 × 4 grid is 16 unit squares of area 1 each, so the whole board's area is 16.
A rectangle's area is just rows × columns of unit squares — that is the Grade 3 definition of area by multiplication.
3.MD.C.7Identify SubproblemsFind the center square's area
The middle 2 × 2 block sits entirely inside the star, a fully shaded square of area 4.
Counting the 4 unit squares in the center is another Grade 3 "area by tiling" move.
3.MD.C.7Draw A DiagramFind the eight points' area
Each of the 8 star points is a right triangle that is half a unit square, so together they add 8 × = 4.
Multiplying the unit fraction 1/2 by the whole number 8 is Grade 4 "fraction times a whole number" arithmetic.
4.NF.B.4Identify SubproblemsAdd the two areas
Add the pieces: center square (4) plus the eight points (4) give a star area of 8.
Adding the areas of non-overlapping pieces to get the whole shape's area is the Grade 3 area-as-addition idea.
The eight-pointed star covers exactly 8 of the grid's 16 unit squares.
▸ Why?
The star has no gaps or overlaps inside it, so its total area is just the areas of its pieces added together: the central square plus the eight points.
▸ Why?
The central piece is a full 2-by-2 block of the grid, which holds 2 rows of 2 unit squares, so it contributes an area of 4.
▸ Why?
Two rows of two unit squares is two equal groups of two, and that counts up to 4.
▸ Why?
The eight points contribute an area of 4, because each point is exactly half of one unit square and eight halves add up to 4.
▸ Why?
Each point is a unit square cut along its diagonal, and that diagonal splits the square into two matching pieces, so one point is half a square's area.
▸ Why?
Flipping one half of the cut square lays it exactly onto the other half, so the two halves have equal area and each is half the whole square.
▸ Why?
Eight points of one-half each is eight equal groups of a half, and that adds up to 4 whole units.
Convert to a percent
The star covers 8 of the 16 cells: = = 50%, matching choice (B).
Expressing a part-to-whole ratio as a percent is Grade 6 ratio and percent reasoning.
6.RP.A.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 percent and ratio reasoning you already know — once you split the star into a center square and eight half-square points!
- Find the board's total area
- Find the center square's area
- Find the eight points' area
- Add the two areas
- Convert to a percent
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 2000 #25 Gr 6Similar level
Same sub-type: Polygon TriangulationSame techniqueArea Difference Identify SubproblemsThe area of rectangle ABCD is 72 units squared. If point A and the midpoints of BC and CD are joined to form a triangle,…
- AMC 8 2007 #23 Gr 6Similar level
Same sub-type: Polygon TriangulationSame techniqueArea Difference Identify SubproblemsWhat is the area of the shaded pinwheel shown in the 5 × 5 grid?
- AMC 8 2008 #23 Gr 6Similar level
Same sub-type: Polygon TriangulationSame techniqueArea Difference Identify SubproblemsIn square ABCE, AF=2FE and CD=2DE. What is the ratio of the area of △ BFD to the area of square ABCE?