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AMC 8 · 2025 · #1

Grade 6 geometry-2d
Archetype Compound Area by Decomposition / Difference
area-rectanglesarea-trianglesfraction-arithmeticpercentage identify-subproblemsarea-difference ↑ Prerequisites: area-rectanglesfraction-arithmetic
📏 Medium solution 💡 2 insights 📊 Diagram

Problem

The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire 4×44\times4 grid is covered by the star?

Pick an answer.

(A)
40
(B)
50
(C)
60
(D)
75
(E)
80
View mode:

Toolkit + CCSS Solution

Understand

Restated: An eight-pointed star is drawn inside a $4 \times 4$ unit grid. What percent of the grid's area is shaded by the star?

Givens: The whole figure sits on a $4 \times 4$ grid, so there are $16$ unit squares of total area; The star is built from a fully shaded central $2 \times 2$ square together with $8$ shaded triangular "points"; Each star point is a right triangle that occupies exactly half of a unit square (leg lengths $1$ and $1$, or base $2$ and height $1$ split in two); Answer choices: (A) $40$, (B) $50$, (C) $60$, (D) $75$, (E) $80$ (percent)

Unknowns: The shaded star area as a percent of the total $4 \times 4$ grid area

Understand

Restated: An eight-pointed star is drawn inside a $4 \times 4$ unit grid. What percent of the grid's area is shaded by the star?

Givens: The whole figure sits on a $4 \times 4$ grid, so there are $16$ unit squares of total area; The star is built from a fully shaded central $2 \times 2$ square together with $8$ shaded triangular "points"; Each star point is a right triangle that occupies exactly half of a unit square (leg lengths $1$ and $1$, or base $2$ and height $1$ split in two); Answer choices: (A) $40$, (B) $50$, (C) $60$, (D) $75$, (E) $80$ (percent)

Plan

Primary tool: #7 Identify Subproblems

Secondary: #1 Draw a Diagram, #3 Eliminate Possibilities

The star is a single complicated shape, but it is built from simple pieces that already line up with the grid: one central $2 \times 2$ 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).

Execute — Answer: B

#7 Identify Subproblems 3.MD.C.7 Step 1
  • Compute the total area of the board.
  • The $4 \times 4$ grid is made of $16$ unit squares, each of area $1$, so the whole grid has area $16$ square units.
$$4 \times 4 = 16$$

💡 A rectangle's area is just rows $\times$ columns of unit squares — that is the Grade 3 definition of area by multiplication.

#1 Draw a Diagram 3.MD.C.7 Step 2
  • Mark the fully shaded center.
  • The middle $2 \times 2$ block of the grid is completely inside the star, so it contributes a square region of side $2$.
$$2 \times 2 = 4$$

💡 Counting the $4$ unit squares in the center is another Grade 3 "area by tiling" move.

#7 Identify Subproblems 4.NF.B.4 Step 3
  • Each of the $8$ "points" of the star is a right triangle whose two legs lie along grid lines with lengths $1$ and $1$ — exactly half of a unit square.
  • So each point has area $\tfrac{1}{2}$, and the eight points together contribute $8 \times \tfrac{1}{2} = 4$.
$$8 \times \tfrac{1}{2} = 4$$

💡 Multiplying the unit fraction $\tfrac{1}{2}$ by the whole number $8$ is Grade 4 "fraction times a whole number" arithmetic.

#7 Identify Subproblems 3.MD.C.7 Step 4
  • Add the subproblem answers to get the star's total area.
  • Center square ($4$) plus eight points ($4$) gives a star area of $8$ square units.
$$4 + 4 = 8$$

💡 Adding the areas of non-overlapping pieces to get the whole shape's area is the Grade 3 area-as-addition idea.

#3 Eliminate Possibilities 6.RP.A.3 Step 5
  • Form the fraction of the grid covered by the star and convert it to a percent.
  • The star covers $8$ out of $16$ unit squares, which is $\tfrac{8}{16} = \tfrac{1}{2}$, and one half is $50\%$.
  • Tool #3 (Eliminate Possibilities) confirms the answer is choice (B).
$$\dfrac{8}{16} = \dfrac{1}{2} = 50\% \;\Rightarrow\; \textbf{(B)}$$

💡 Expressing a part-to-whole ratio as a percent is Grade 6 ratio and percent reasoning.

[1] #7 3.MD.C.7 Compute the total area of the board. The $4 \times 4$ grid is made of $16$ unit
[2] #1 3.MD.C.7 Mark the fully shaded center. The middle $2 \times 2$ block of the grid is compl
[3] #7 4.NF.B.4 Each of the $8$ "points" of the star is a right triangle whose two legs lie alon
[4] #7 3.MD.C.7 Add the subproblem answers to get the star's total area. Center square ($4$) plu
[5] #3 6.RP.A.3 Form the fraction of the grid covered by the star and convert it to a percent. T

Review

Reasonableness: Cross-check by the complement: the unshaded region is made of $4$ corner triangles (each a right triangle with legs $2$ and $1$, so area $\tfrac{1}{2} \cdot 2 \cdot 1 = 1$, total $4$) plus $4$ unit squares tucked next to the corners (total $4$). The unshaded area is $4 + 4 = 8$, so the shaded star area is $16 - 8 = 8$ — the same $50\%$. Also, by symmetry the star looks like it covers "about half" of the grid, so $50\%$ is the only believable choice; $40\%$ would be too sparse and $60\%/75\%/80\%$ would visibly engulf the corners.

Alternative: Tool #16 (Change Focus / Count the Complement): instead of summing the star pieces, count the unshaded pieces directly — four corner triangles of area $1$ each and four interior unit squares of area $1$ each give $8$ unshaded out of $16$, so shaded $= 16 - 8 = 8$, giving the same $50\%$. This is often faster when the "background" pieces are simpler than the figure itself.

CCSS standards used (min grade 6)

  • 3.MD.C.7 Relate area to multiplication and addition operations (Computing the $4 \times 4 = 16$ total grid area, the $2 \times 2 = 4$ central-square area, and adding the central square to the eight points to get the star area $4 + 4 = 8$.)
  • 4.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a whole number (Multiplying the half-square point area $\tfrac{1}{2}$ by the number of points $8$ to get $8 \times \tfrac{1}{2} = 4$.)
  • 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Converting the part-to-whole ratio $\tfrac{8}{16} = \tfrac{1}{2}$ into the percent $50\%$ that the answer choices use.)

⭐ This 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!

⭐ This 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!

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