2024 AMC 8 Problem 3

One Page to Learn. One Point to Earn.

In This Page:

To help you master 2024 AMC 8 Problem 3, 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.

2024 AMC8 Problem 3

Solution

Solution 1

Area Subtraction Method

The visible part of any square is simply its total area minus the area of the square that is stacked directly on top of it.

The visible area of a flat shape partially covered by another shape is the area of the bottom shape minus the area of the top shape.

Step 1: Calculate the area of each square

Since the side lengths are given, square each side length to find the total area of each square as if it were completely visible. Let $A_1$ be the area of the smallest square, up to $A_4$ for the largest square.

$$A_1=4^2=16$$

$$A_2=7^2=49$$

$$A_3=9^2=81$$

$$A_4=10^2=100$$

Step 2: Calculate the visible area of the first gray region

The first gray square has a total area of 49. However, the smaller white square is placed directly on top of it, covering an area of 16. Subtract the covered area from the total area of the gray square.

$${\text{Visible Gray}}_1=A_2-A_1$$

$${\text{Visible Gray}}_1=49-16=33$$

Step 3: Calculate the visible area of the second gray region

Similarly, the largest gray square has an area of 100. It is partially covered by the second white square, which has an area of 81. Subtract this covered area.

$${\text{Visible Gray}}_2=A_4-A_3$$

$${\text{Visible Gray}}_2=100-81=19$$

Step 4: Find the total visible gray area

Add the visible areas of the two separate gray regions together to find the final answer.

$$\text{Total Gray Area}=33+19=52$$

Correct answer: E. $52$
The total visible gray area is $52$, which matches answer choice E. We can also check the order of magnitude: the maximum possible area is 100, and visually roughly half the area appears gray, making 52 a highly reasonable result.

Solution 2

Complementary Area Method

Instead of calculating the gray regions directly, we can find the total area of the white regions and subtract that from the area of the entire figure.

Step 1: Identify the total area of the figure

Because all the squares are aligned at the bottom-left corner and stacked in increasing order, the entire shape fits exactly inside the largest square.

$$\text{Total Area}=10^2=100$$

Step 2: Calculate the area of each visible white region

The first white region is just the smallest square, which is fully visible. The second white region is the third square minus the area covered by the second square.

$${\text{Visible White}}_1=4^2=16$$

$${\text{Visible White}}_2=9^2-7^2=81-49=32$$

Step 3: Find the total visible white area

Add the areas of the two visible white regions together.

$$\text{Total White Area}=16+32=48$$

Step 4: Subtract the white area from the total area

Every part of the largest square is either visible white or visible gray. By subtracting the total white area from the total area of the largest square, we find the remaining gray area.

$$\text{Total Gray Area}=100-48=52$$

Correct answer: E. $52$
The total visible gray area is $52$. This complementary method gives the exact same result as our direct area subtraction method, confirming our arithmetic is correct.

Solution 3

Rectangle Decomposition Method

Any L-shaped polygon can be split into two non-overlapping rectangles. We can slice the gray regions into smaller rectangles and add their areas together.

Step 1: Decompose the first gray band

Slice the inner gray band (from the square of side 7) into two pieces by extending the right edge of the small white square upwards. This creates a vertical rectangle on the right and a horizontal rectangle on the top.

Step 2: Calculate the rectangles for the first gray band

The right rectangle has a width equal to the difference in side lengths of the two squares, and a height equal to the smaller square’s side. The top rectangle has a width equal to the larger square’s side, and a height equal to the difference in side lengths.

$${\text{Right Rectangle}}_1=(7-4)\times 4=3\times 4=12$$

$${\text{Top Rectangle}}_1=7\times (7-4)=7\times 3=21$$

Step 3: Decompose and calculate the second gray band

Slice the outer gray band (from the square of side 10) in the exact same way, using the side lengths of the third and fourth squares.

$${\text{Right Rectangle}}_2=(10-9)\times 9=1\times 9=9$$

$${\text{Top Rectangle}}_2=10\times (10-9)=10\times 1=10$$

Step 4: Sum the areas of the rectangles

Add the areas of all four rectangles together to find the total visible gray area.

$$\text{Total Gray Area}=12+21+9+10=52$$

Correct answer: E. $52$
Summing the rectangular pieces gives $\boxed{52}$. This perfectly matches our subtraction methods. This confirms our geometric interpretation of the L-shapes is accurate.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2024 Question 3. 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 1

2025 AMC 8 Problem 12

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 Square:

$A$ is the area of the square, and $s$ is the side length. Used to find the total possible area of each of the four overlapping squares.

• Area Subtraction Principle:

Applies when a smaller shape is placed entirely inside or on top of a larger shape. Because the squares are aligned at the same corner, each smaller square is completely contained within the larger squares beneath it.

• Decomposition of Polygons:

The parts must not overlap and must completely cover the desired region. The L-shaped visible gray regions are not basic rectangles, but they can be sliced into standard rectangular pieces whose areas are easy to calculate.

Key Takeaway: Remember This Strategy for Next Time

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!