This is a square centered at the origin with vertices at $ (\pm4, 0), (0, \pm4) $. The area is:

["# The Square at the Origin with Vertices at $ (\pm4, 0), (0, \pm4) $: What Is Its Area?", "When you visualize a square centered at the origin with vertices precisely at $ (4, 0), (-4, 0), (0, 4), $ and $ (0, -4) $, it creates a clear geometric figure starring symmetry and precision. This square offers a perfect chance to explore its fundamental properties—especially its area—using simple geometry.", "## Understanding the Geometry", "The given square has vertices located symmetrically on the axes:", "- Right vertex: $ (4, 0) $\n- Left vertex: $ (-4, 0) $\n- Top vertex: $ (0, 4) $\n- Bottom vertex: $ (0, -4) $", "Plotting these points reveals that the square’s sides lie along the lines connecting these axis points, forming a diamond (rotated square) aligned with the coordinate axes.", "The key insight: the side length is the distance between adjacent vertices along the axes. For example, the distance from $ (4, 0) $ to $ (0, 4) $ is not a side but determined via the diagonals.", "Actually, in this configuration, the diagonals run horizontally and vertically, stretching from $ (-4, 0) $ to $ (4, 0) $ (horizontal diagonal) and from $ (0, -4) $ to $ (0, 4) $ (vertical diagonal).", "- Length of the horizontal diagonal: $ 8 $ units\n- Length of the vertical diagonal: $ 8 $ units", "### Diagonals and Area Formula", "For any rhombus or square, the area can be calculated using the diagonals:", "$$\n\ ext{Area} = \frac{1}{2} \ imes d_1 \ imes d_2\n$$", "where $ d_1 $ and $ d_2 $ are the lengths of the diagonals.", "Here:", "$$\n\ ext{Area} = \frac{1}{2} \ imes 8 \ imes 8 = \frac{1}{2} \ imes 64 = 32\n$$", "## Final Answer", "The area of the square (or diamond) centered at the origin with vertices at $ (\pm4, 0) $ and $ (0, \pm4) $ is:", "$$\n\boxed{32}\n$$", "## Why This Formula Works", "Even though the shape is described as a square, the use of diagonal lengths ensures accuracy—especially when vertices lie on axes or diagonals. While the side length here can be computed as $ 4\sqrt{2} $ from the distance between $ (4, 0) $ and $ (0, 4) $, the diagonal-based area formula simplifies straightforward computation without needing trigonometry or coordinate transformations.", "---", "This elegant configuration demonstrates how symmetry simplifies geometric analysis—ideal for teaching coordinate geometry, graphical transformations, or understanding area formulas in non-standard polygons."]