 Aaliyah owns a farm with a goat. This goat will eat anything in sight, especially grass, so she keeps it
tied to a 15-foot leash that is tethered to the corner of a square barn with 10-foot side length.
Assuming the barn is surrounded by grass and the goat cannot enter the building, what is the square
footage of grass within the goat’s reach?
[Note the area of a circle is given by the formula 𝐴 = 𝜋𝑟2 ,
where 𝜋 ≈ 3.14159 and 𝑟 is the radius of that circle]


To find the total square footage of grass within the goat's reach, we need to break the problem down into two parts. First, let's consider the area of the full circle that the goat could cover if there were no barn, and then account for the barn, which limits the area.

### Step 1: Compute the full circle area
The goat is tied to a 15-foot leash, so if there were no barn, the area within the goat's reach would be a full circle with a radius of 15 feet. The formula for the area of a circle is:

\[
A = \pi r^2
\]

Using \( r = 15 \) feet:

\[
A_{\text{circle}} = \pi \times 15^2 = \pi \times 225 \approx 3.14159 \times 225 = 706.86 \text{ square feet}.
\]

This is the total area if the goat had no obstacles, but we need to adjust for the fact that part of the circle is blocked by the barn.

### Step 2: Account for the barn
The barn is a square with a side length of 10 feet, and the goat is tied to one corner of the barn. The barn restricts the goat's movement in two perpendicular directions. Since the barn covers part of the circle, we need to compute the area of grass available in three distinct sections.

1. **Quarter-circle in one quadrant**: In one quadrant (with respect to the corner), the goat can cover a quarter of the circle with radius 15 feet.
   
   The area of a quarter-circle is:
   \[
   A_{\text{quarter-circle}} = \frac{1}{4} \times \pi \times r^2 = \frac{1}{4} \times \pi \times 15^2 = \frac{1}{4} \times 706.86 = 176.715 \text{ square feet}.
   \]

2. **Circular segment in another quadrant**: In the second quadrant, the barn limits the radius. The distance from the barn corner to the wall is 10 feet, so in this quadrant, the goat can only cover part of the circle with radius 10 feet.

   The area of this quarter-circle with radius 10 feet is:
   \[
   A_{\text{quarter-circle with radius 10}} = \frac{1}{4} \times \pi \times 10^2 = \frac{1}{4} \times \pi \times 100 = \frac{1}{4} \times 314.159 = 78.54 \text{ square feet}.
   \]

3. **Circular segment in another quadrant**: The goat can move along the outer edge of the barn in the remaining quadrant, creating a quarter-circle segment with radius \( 15 - 10 = 5 \) feet. This forms a smaller segment.

   The area of this quarter-circle is:
   \[
   A_{\text{quarter-circle with radius 5}} = \frac{1}{4} \times \pi \times 5^2 = \frac{1}{4} \times \pi \times 25 = \frac{1}{4} \times 78.54 = 19.63 \text{ square feet}.
   \]

### Step 3: Total reachable area
To find the total square footage of grass within the goat's reach, sum the areas of all these sections:

\[
A_{\text{total}} = A_{\text{quarter-circle}} + A_{\text{quarter-circle with radius 10}} + A_{\text{quarter-circle with radius 5}} = 176.715 + 78.54 + 19.63 = 274.885 \text{ square feet}.
\]

Thus, the goat can reach approximately **274.89 square feet** of grass.

---

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What would the reachable area be if the barn was a rectangle instead of a square?
2. How would the solution change if the goat's leash was 20 feet instead of 15?
3. What happens to the grass area if the barn is not tethered to the corner but along the side?
4. Can we calculate the perimeter of the area the goat can graze using a similar method?
5. How would the calculation change if the barn had rounded corners?

**Tip:** When solving geometry problems with obstacles, try visualizing the situation and breaking it into manageable parts like circles, segments, or quadrants.

Aaliyah owns a farm with a goat. This goat will eat anything in sight, especially grass, so she keeps it tied to a 15-foot leash that is tethered to the corner of a square barn with 10-foot side length. Assuming the barn is surrounded by grass and the goat cannot enter the building, what is the square footage of grass within the goat’s reach?

Explore how to calculate the square footage of grass a goat can reach when tied to a 15-foot leash attached to the corner of a square barn. This problem involves key geometry concepts such as area of a circle, circular segments, and quarter-circles. Perfect for students in Grades 9-12.

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