1. **Problem Statement:** Find the yellow area inside a square of side length 8, below the triangle formed by two radii of the inscribed circle meeting the bottom corners of the square. 2. **Understanding the figure:** - The square has side length $8$. - The inscribed circle touches all sides, so its radius $r$ is half the side length: $r = \frac{8}{2} = 4$. - The circle's center is at the center of the square. - The two radii to the bottom corners form an isosceles triangle with base $8$ and two sides equal to the radius $4$. 3. **Calculate the area of the sector formed by the two radii:** - The angle $\theta$ between the two radii can be found using the cosine rule in the triangle formed by the two radii and the base: $$\cos(\theta) = \frac{4^2 + 4^2 - 8^2}{2 \times 4 \times 4} = \frac{16 + 16 - 64}{32} = \frac{-32}{32} = -1$$ - So, $\theta = \pi$ radians (180 degrees). - The sector area is: $$\text{Sector area} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{\pi}{2\pi} \times \pi \times 4^2 = \frac{1}{2} \times \pi \times 16 = 8\pi$$ 4. **Calculate the area of the triangle formed by the two radii and the chord (the base):** - The triangle is isosceles with sides $4,4$ and base $8$. - Using the formula for area with base and height: - The height $h$ is found by Pythagoras: $$h = \sqrt{4^2 - (8/2)^2} = \sqrt{16 - 16} = 0$$ - The height is zero, meaning the two radii and the base form a straight line, so the triangle area is 0. 5. **Calculate the area of the segment (yellow area):** - The yellow area is the area of the sector minus the triangle area: $$\text{Yellow area} = 8\pi - 0 = 8\pi$$ 6. **Final answer:** $$\boxed{8\pi}$$ This is the yellow area inside the square below the triangle formed by the two radii to the bottom corners.