2025 AMC 8 Problem 10

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To help you master 2025 AMC 8 Problem 10, 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

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2025 AMC8 Problem 10

Solution

Solution 1

Principle of Inclusion-Exclusion Method

The total area of two overlapping shapes is the sum of their individual areas minus the area of the region they both share (the overlap).

The Principle of Inclusion-Exclusion for area states that $\text{Area}(A\cup B)=\text{Area}(A)+\text{Area}(B)-\text{Area}(A\cap B)$.

Step 1: Identify the extents of the original rectangle

The original rectangle has a width of 5 and a height of 3. The rotation point M is the midpoint of the bottom edge. From M, the original rectangle extends 2.5 inches left, 2.5 inches right, 3 inches up, and 0 inches down.

$$\text{Original Extents:}$$

$$\text{Left}=2.5,\text{Right}=2.5$$

$$\text{Up}=3,\text{Down}=0$$

Step 2: Determine the extents of the rotated rectangle

When the rectangle is rotated $90^{\circ }$ clockwise around M, the directions from M rotate as well. What was ‘Up’ rotates to become ‘Right’, ‘Left’ becomes ‘Up’, ‘Right’ becomes ‘Down’, and ‘Down’ becomes ‘Left’. We apply this to the original extents.

$$\text{Rotated Extents:}$$

$$\text{New Right (from Old Up)}=3$$

$$\text{New Up (from Old Left)}=2.5$$

$$\text{New Down (from Old Right)}=2.5$$

$$\text{New Left (from Old Down)}=0$$

Step 3: Find the dimensions of the overlapping region

The overlap is the region that belongs to both rectangles. Its boundary in any direction is the smaller of the two rectangles’ extents in that direction. The overlap extends 0 to the left (min of 2.5 and 0), 2.5 to the right (min of 2.5 and 3), 2.5 up (min of 3 and 2.5), and 0 down (min of 0 and 2.5). This forms a square.

$$\text{Overlap Width}=2.5-0=2.5$$

$$\text{Overlap Height}=2.5-0=2.5$$

$$\text{Area of Overlap}=2.5\times 2.5=6.25$$

Step 4: Calculate the total combined area

The total area is the sum of the areas of the two rectangles minus the area of their overlap.

$$\text{Area of each rectangle}=5\times 3=15$$

$$\text{Total Area}=15+15-6.25=23.75$$

Correct answer: D. $23.75$
The total area is 23.75 square inches, which corresponds to answer choice D. We can verify this by checking that the overlap is entirely contained within both bounding boxes, which logically restricts its maximum possible area to less than 15.

Solution 2

Non-Overlapping Regions Method (Dissection)

Instead of subtracting the overlap, we can start with the entire area of the original rectangle and add the pieces of the rotated rectangle that stick out from it.

Area Addition Postulate: The area of a geometric shape is the sum of the areas of its non-overlapping parts.

Step 1: Calculate the area of the original rectangle

We will use the original rectangle as our base shape. Its area is simply its width multiplied by its height.

$${\text{Area}}_{\text{Original}}=5\times 3=15$$

Step 2: Calculate the area of the bottom extension

The rotated rectangle extends 2.5 inches below the rotation point M, while the original rectangle does not go below M at all. This new lower section has a width of 3 inches (extending from 0 on the left to 3 on the right) and a height of 2.5 inches.

$${\text{Area}}_{\text{Bottom}}=3\times 2.5=7.5$$

Step 3: Calculate the area of the right extension

Above the horizontal line through M, the rotated rectangle extends 3 inches to the right, but the original rectangle ends at 2.5 inches to the right. This creates an extra vertical strip of width 0.5 inches. The height of this strip matches the upward extent of the rotated rectangle, which is 2.5 inches.

$${\text{Area}}_{\text{Right}}=(3-2.5)\times 2.5=0.5\times 2.5=1.25$$

Step 4: Sum the non-overlapping parts

The total area is the sum of the original rectangle and the two newly calculated extensions. (Note that the rotated rectangle does not extend beyond the original shape to the left or top).

$$\text{Total Area}=15+7.5+1.25=23.75$$

Correct answer: D. $23.75$
The total area is 23.75. This perfectly matches the result from Method 1, confirming our geometric logic.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2025 Question 10. These real AMC 8 problems test the same key concept—Area. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2022 AMC 8 Problem 4

2006 AMC 8 Problem 4

2026 AMC 8 Problem 6

2026 AMC 8 Problem 13

2025 AMC 8 Problem 1

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Principle of Inclusion-Exclusion (Area):

$A$ and $B$ are any two overlapping shapes. $A\cup B$ is the total combined figure, and $A\cap B$ is the overlapping region. Adding the area of the two individual rectangles counts the overlapping section twice. Subtracting the overlap once corrects this double-counting.

• Effect of 90-Degree Rotations on Dimensions:

Applies to orthogonal boundaries (horizontal and vertical lines) relative to a specific center of rotation. Since the original rectangles are aligned horizontally and vertically, this simple mapping allows us to find the exact boundaries of the rotated rectangle without using complex coordinate geometry.

Key Takeaway: Remember This Strategy for Next Time

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