2025 AMC 8 Problem 1
One Page to Learn. One Point to Earn.
In This Page:
To help you master 2025 AMC 8 Problem 1, this page includes:
- The original AMC problem to try, plus the correct answer
- Clear, step-by-step solutions using different methods
- Key ideas, useful formulas, and takeaways you can apply to similar problems
- Related problems from past AMC exams, plus new practice problems that build the same skills
Want more practice? Scroll down to download Think Academy’s AMC 8 practice packs and keep building your AMC 8 skills.
2025 AMC8 Problem 1
1.
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire grid is covered by the star?
Solution
Geometric Decomposition Method
Break the star into standard geometric shapes (a large central square and four outer triangles) whose areas can be calculated directly.
Triangle Area Formula: The area of a triangle is , where is the base and is the perpendicular height. In grid geometry, look for triangles whose bases and heights align with the grid lines.
Step 1: Identify the total area
Assume each small square in the grid has an area of 1 unit. Since the grid is 4 squares wide and 4 squares tall, we find the total area by multiplying its dimensions.
Step 2: Calculate the area of the central body
The main core of the star perfectly covers a square block in the middle of the grid. This block is 2 squares wide and 2 squares tall.
Step 3: Calculate the area of the four star points
The star has four points extending outward. Each point is a triangle whose base spans the width of the central block and whose height extends to the edge of the grid. We use the triangle area formula to find the area of one point, then multiply by 4 to get the total area for all four points.
Step 4: Determine the total percentage
We add the area of the center and the area of the points to find the total covered area. Finally, we divide this by the total grid area and convert it to a percentage.
The exact answer is , corresponding to choice B. We can verify this by calculating the unshaded area instead: 4 empty corner squares plus 8 half-empty edge squares gives 4 + 4 = 8 unshaded squares. Since 16 – 8 = 8, the shaded area is indeed exactly half the grid.
Square-by-Square Classification Method
Instead of looking at the star as a whole, look at how much of each individual small square is shaded, grouping similar squares together.
Classification Counting: When a total is made up of many small parts, grouping them by type (e.g., center, edges, corners) ensures you don’t miss any pieces and makes calculations simpler.
Step 1: Classify the squares in the grid
The grid consists of 16 small squares. We can group them based on their location: 4 in the exact center, 8 along the edges, and 4 at the very corners.
Step 2: Count the covered area in the center
The 4 squares that make up the center of the grid are entirely filled by the star pattern.
Step 3: Count the covered area in the edge squares
There are 8 squares along the outer edges that are not corners. In each of these squares, the star’s boundary runs exactly along the diagonal, filling exactly half of the square.
Step 4: Count the covered area in the corner squares
The 4 squares at the corners of the grid are completely white and contain no part of the star.
Step 5: Determine the total percentage
We add the shaded contributions from the center, edges, and corners to find the total covered area, and then calculate what percent this is of the entire 16-square grid.
The total area is 8, which is 50% of 16. This perfectly matches the result from Method 1, confirming choice B.
Visual Rearrangement (Cut-and-Paste) Method
Rearrange the shaded parts mentally to form full squares, making the fraction of covered area obvious without any numerical formulas.
Conservation of Area: Cutting a shape and moving its pieces to new locations does not change its total area. This cut-and-paste method is a powerful tool for visual problem solving.
Step 1: Notice the complementary shapes
Observe the unshaded regions. The 4 corner squares are entirely empty. Meanwhile, the 8 edge squares each contain a shaded triangle that covers exactly half of the square.
Step 2: Pair up the half-squares
Imagine cutting out the 8 shaded triangular half-squares from the edges of the star. Since two identical halves make a whole, we can combine them to form full squares.
Step 3: Relocate the shaded pieces visually
We can mentally move these 4 newly formed full squares into the 4 empty corner squares of the grid. They will fit perfectly.
Step 4: Determine the total percentage
After this rearrangement, the 4 center squares and the 4 corner squares are entirely shaded, while the rest of the grid is completely empty. The shaded squares represent exactly half of the total grid.
Without doing complex arithmetic, we have filled exactly 8 whole squares out of 16. This yields percent (choice B) efficiently and confirms our previous methods.
Similar Problems in Real Exams
The following five AMC 8 problems are closely related to AMC 8 2025 Question 1. These real AMC 8 problems test the same key concept—Composite Area & Geometric Counting. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.
2026 AMC 8 Problem 6
2025 AMC 8 Problem 12
2024 AMC 8 Problem 3
2020 AMC 8 Problem 24
2019 AMC 8 Problem 4
Adapted Problems to Practice
What Formulas Do You Need for This Problem?
- Area of a Triangle:
A is the area, b is the base length, and h is the perpendicular height. Used in Method 1 to find the area of the 4 triangular points of the star.
- Area Addition Postulate (Decomposition):
The area of a complex figure is equal to the sum of the areas of its non-overlapping parts.
Applies to any shape that can be split into simpler recognizable geometric pieces. Allows us to calculate the star's total area by summing the areas of the central square and the outer triangles, or by summing individual grid squares.
- Conservation of Area (Visual Rearrangement):
Cutting a shape into pieces and relocating those pieces does not change the total area.
Highly effective on grids when shapes (like half-squares) can perfectly fill empty spaces (like whole squares). Used in Method 3 to move the 8 shaded half-squares into the 4 empty corner squares, making the 50% ratio visually obvious without calculation.
Key Takeaway: Remember This Strategy for Next Time
A complex figure drawn on a square grid with vertices on grid lines.
Try: Assign an area of 1 to each small grid square and break the complex figure down into full squares and simple triangles.
Why: Grid lines provide straightforward bases and heights, making basic area formulas easy to apply.
Slanted boundary lines that go from one corner of a grid square exactly to the opposite corner.
Try: Count that square as exactly half shaded.
Why: The diagonal of a square perfectly divides its area into two equal right triangles.
Many identical "fractional" pieces (like half-squares) alongside empty whole spaces.
Try: Use the cut-and-paste method to mentally move the fractional pieces into the empty spaces to form whole squares.
Why: Rearranging pieces preserves total area while transforming a complex shape into a simple rectangle or a recognizable block of squares.
Unlock More Targeted AMC 8 Practice Resource
Build Stronger Skills. Practice with Purpose. Prepare with Confidence.
Get a complete set of module-focused AMC 8 practice workbooks designed to help students strengthen core concepts, improve problem-solving strategies, and feel more confident on test day.
With 250 practice problems organized across six essential AMC 8 modules, students can focus on one skill area at a time and learn from clear, step-by-step solutions.
Download the Free AMC 8 Practice Resource Pack
Inside, You’ll Find:
- 6 module-focused practice workbooks: Algebra, Geometry, Number Theory, Combinatorics, and Probability & Statistics.
- 250 AMC 8-style practice problems covering essential concepts and common competition question types.
- Complete answer keys and step-by-step solutions to help students understand the reasoning behind each answer.
- Organized, print-ready materials for independent study, guided practice, or extra preparation at home.
Get Instant Access: Fill out the form below to receive our exclusive resource pack.
*Please enter a valid email address to access the materials and receive future prep updates.
About Think Academy
Think Academy US Online, part of TAL Education Group (NYSE: TAL), provides structured math and ELA programs that help students build strong academic foundations and grow with confidence. With expert teachers, proven methods, and innovative AI tools, we nurture learning ability, curiosity, and healthy study habits that support long-term growth and a lifelong love of learning.
Check our programs:
- Think Kids: Ages 3-5
- Math
- Lower Elementary Math — Grades 1–2
- Upper Elementary Math — Grades 3–5
- Middle School Math — Grade 6–8
- High School Math — Grade 9-12
- ELA
- Elementary ELA — Grades 2–5
Want more insights on Math & ELA learning and parenting? Fill out the form on this page and subscribe to our newsletter for weekly tips and the latest resources. Join 100,000+ families in our Parent Community!
Leave a Comment & Share Your Thoughts!
Share This Story, Choose Your Platform!
Related Posts
