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QUESTION: If the center of a hexagon is located at a (0,0) coordinate, which are the coordinates of each vertex if the length of the side of the hexagon is 10 mm?

Analyzing and reasoning phase

Okay, so the user wants to find the coordinates of each vertex of a hexagon centered at (0,0) with a side length of 10 mm. Let me think about how to approach this. First, I remember that a regular hexagon can be inscribed in a circle. Each vertex is equally spaced around the circumference. The key here is relating the side length to the radius of the circumscribed circle. Wait, is the radius the same as the side length? Hmm, no, I think that's only true for a hexagon with a side length equal to the radius. Wait, actually, in a regular hexagon, the side length is equal to the radius of the circumscribed circle. Let me confirm that. In a regular hexagon, all sides are equal, and all internal angles are 120 degrees. The radius of the circumscribed circle (distance from center to a vertex) is equal to the side length. So if the side length is 10 mm, the radius is also 10 mm. So each vertex is 10 mm away from the center. Now, to find the coordinates of the six vertices. Since it's centered at (0,0), I can use polar coordinates to Cartesian coordinates conversion. Each vertex will be at an angle of 60 degrees apart, starting from some initial angle. Typically, one vertex is along the positive x-axis, and then they go counterclockwise from there. So the angles would be 0°, 60°, 120°, 180°, 240°, 300°, right? Then converting each polar coordinate (10 mm, angle) to Cartesian coordinates (x,y). The formula for conversion is: x = r * cos(theta) y = r * sin(theta) But I need to make sure the angles are in radians if using a calculator, but since we're doing it symbolically, we can keep them in degrees. Let me list each vertex: 1. 0°: (10*cos(0°), 10*sin(0°)) = (10, 0) 2. 60°: (10*cos(60°), 10*sin(60°)) = (10*(1/2), 10*(√3/2)) = (5, 5√3) 3. 120°: (10*cos(120°), 10*sin(120°)) = (10*(-1/2), 10*(√3/2)) = (-5, 5√3) 4. 180°: (10*cos(180°), 10*sin(180°)) = (-10, 0) 5. 240°: (10*cos(240°), 10*sin(240°)) = (10*(-1/2), 10*(-√3/2)) = (-5, -5√3) 6. 300°: (10*cos(300°), 10*sin(300°)) = (10*(1/2), 10*(-√3/2)) = (5, -5√3) Wait, let me check the cosine and sine values for each angle. For 0°, that's straightforward. 60°, cos is 0.5, sin is √3/2. 120°, cosine is -0.5, sine is √3/2. 180°, cosine is -1, sine 0. 240°, which is 180+60, cosine -0.5, sine -√3/2. 300°, which is 360-60, cosine 0.5, sine -√3/2. Yes, that seems right. So the coordinates are: (10, 0), (5, 5√3), (-5, 5√3), (-10, 0), (-5, -5√3), (5, -5√3). Then back to the starting point. But wait, I need to confirm that the side length is indeed equal to the radius. Wait, maybe I made a mistake here. Because in a regular hexagon, the radius (distance from center to vertex) is equal to the side length. Let me verify that. Yes, in a regular hexagon, the radius of the circumscribed circle is equal to the length of each side. So if the side length is 10 mm, then the radius is 10 mm. Therefore, the coordinates calculated above should be correct. Alternatively, if the user had provided the distance from the center to a side (the apothem), the calculation would be different. But here, the side length is given, so the radius is the same as the side length. Therefore, the coordinates are as calculated. So the six vertices are those six points. Let me just list them in order, perhaps starting from the rightmost point (10,0) and going counterclockwise. So the coordinates are: 1. $(10, 0)$ 2. $(5, 5\sqrt{3})$ 3. $(-5, 5\sqrt{3})$ 4. $(-10, 0)$ 5. $(-5, -5\sqrt{3})$ 6. $(5, -5\sqrt{3})$ Each subsequent vertex is 60 degrees apart. To confirm, the distance between adjacent vertices should be 10 mm. Let's check between (10,0) and (5,5√3). The distance is sqrt[(10-5)^2 + (0 -5√3)^2] = sqrt[25 + 75] = sqrt[100] =10. Yes, that's correct. Similarly between (5,5√3) and (-5,5√3), the distance is sqrt[(-5-5)^2 + (5√3 -5√3)^2] = sqrt[(-10)^2 +0] =10. Wait, but those two points are separated by 120 degrees, right? Wait, no. Wait, the points (5,5√3) and (-5,5√3) are on the same horizontal line, separated by 10 units in x, so the distance is 10 mm. But that's a horizontal line, but in the hexagon, adjacent vertices should be separated by 60 degrees. Wait, but in this case, those two points are separated by 120 degrees. Wait, maybe I made a mistake in the order. Wait, no. Let me visualize the hexagon. Starting at (10,0), then moving 60 degrees up to (5,5√3), which is in the first quadrant. Then next vertex is (-5,5√3), which is in the second quadrant. Then (-10,0), then (-5,-5√3), then (5,-5√3), back to (10,0). So the distance between (5,5√3) and (-5,5√3) is 10, but that's a horizontal line. But in a regular hexagon, each side is of length 10. Wait, but the distance between (5,5√3) and (-5,5√3) is 10 mm, which would be a horizontal side. But in a regular hexagon, the sides are not horizontal except for the ones at 0° and 180°. Wait, maybe I confused the adjacency. Wait, no. Let's list the vertices in order: 1. (10,0) — rightmost point. 2. Next, moving 60 degrees up: (5,5√3). The edge between (10,0) and (5,5√3) is a side of the hexagon, length 10. 3. Then from (5,5√3) to (-5,5√3): that's a horizontal line left, but wait, that's 10 units long. Wait, but in the hexagon, each side is between adjacent vertices. So the edge from (5,5√3) should be to the next vertex, which is (-5,5√3). But the distance between them is 10? Wait, the distance between (5,5√3) and (-5,5√3) is sqrt[(-5-5)^2 + (5√3 -5√3)^2] = sqrt[(-10)^2 +0] =10. So yes, that's correct. But that seems like a horizontal edge, but in a regular hexagon, only two sides are horizontal (the ones at 0° and 180°). Wait, maybe I'm visualizing it wrong. Let me sketch it mentally. Starting at (10,0), then going up 60 degrees to (5,5√3). Then the next vertex is (-5,5√3). So the edge from (5,5√3) to (-5,5√3) is horizontal, which is 10 mm. Then from (-5,5√3) to (-10,0), which is a diagonal down. Then to (-5,-5√3), then to (5,-5√3), then back to (10,0). Wait, but in a regular hexagon, each internal angle is 120 degrees. So the angles at each vertex are 120 degrees. Let me check the angle at (5,5√3). The edges before and after are (10,0) to (5,5√3) to (-5,5√3). The angle between those two edges at (5,5√3) should be 120 degrees. Let me calculate the vectors. From (5,5√3) to (10,0): the vector is (5, -5√3). From (5,5√3) to (-5,5√3): the vector is (-10, 0). The angle between these two vectors can be found using the dot product. The dot product is (5)(-10) + (-5√3)(0) = -50. The magnitude of the first vector is sqrt(5² + (5√3)²) = sqrt(25 +75) = sqrt(100)=10. The magnitude of the second vector is 10. So the cosine of the angle is (-50)/(10*10) = -0.5. Therefore, the angle is 120 degrees, which is correct. So that checks out. Therefore, the coordinates are correctly calculated. Each side is indeed 10 mm, as verified between the first two points and between the second and third points. So the answer should list all six coordinates as above.

The vertices of a regular hexagon centered at $(0,0)$ with side length $10$ mm are:

$$\begin{array}{rl}{\displaystyle }& {\displaystyle <i><span\; style="color:\; \#2b00fe;">(</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">10</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">0</span></i>}<i><span\; style="color:\; \#2b00fe;">)</span></i><i><span\; style="color:\; \#2b00fe;">,</span></i>}\\ {\displaystyle }& {\displaystyle <i><span\; style="color:\; \#2b00fe;">(</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}\sqrt{\mathrm{<i><span\; style="color:\; \#2b00fe;">3</span></i>}}<i><span\; style="color:\; \#2b00fe;">)</span></i><i><span\; style="color:\; \#2b00fe;">,</span></i>}\\ {\displaystyle }& {\displaystyle <i><span\; style="color:\; \#2b00fe;">(</span></i><i><span\; style="color:\; \#2b00fe;">-</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}\sqrt{\mathrm{<i><span\; style="color:\; \#2b00fe;">3</span></i>}}<i><span\; style="color:\; \#2b00fe;">)</span></i><i><span\; style="color:\; \#2b00fe;">,</span></i>}\\ {\displaystyle }& {\displaystyle <i><span\; style="color:\; \#2b00fe;">(</span></i><i><span\; style="color:\; \#2b00fe;">-</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">10</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">0</span></i>}<i><span\; style="color:\; \#2b00fe;">)</span></i><i><span\; style="color:\; \#2b00fe;">,</span></i>}\\ {\displaystyle }& {\displaystyle <i><span\; style="color:\; \#2b00fe;">(</span></i><i><span\; style="color:\; \#2b00fe;">-</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i><i><span\; style="color:\; \#2b00fe;">-</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}\sqrt{\mathrm{<i><span\; style="color:\; \#2b00fe;">3</span></i>}}<i><span\; style="color:\; \#2b00fe;">)</span></i><i><span\; style="color:\; \#2b00fe;">,</span></i>}\\ {\displaystyle }& {\displaystyle <i><span\; style="color:\; \#2b00fe;">(</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i><i><span\; style="color:\; \#2b00fe;">-</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}\sqrt{\mathrm{<i><span\; style="color:\; \#2b00fe;">3</span></i>}}<i><span\; style="color:\; \#2b00fe;">)</span></i>}\end{array}$$

Derivation:
For a regular hexagon, the radius (distance from center to a vertex) equals the side length. Using polar coordinates $(r,\theta )$ with $r=10$ mm and angles $\theta =0^{\circ },{60}^{\circ },{120}^{\circ },{180}^{\circ },{240}^{\circ },{300}^{\circ }$, convert to Cartesian coordinates $(x,y)$ via:

$$\mathrm{<i><span\; style="color:\; \#2b00fe;">x</span></i>}<i><span\; style="color:\; \#2b00fe;">=</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">r</span></i>}\mathrm{<i><span\; style="color:\; \#2b00fe;">cos</span></i>}<i><span\; style="color:\; \#2b00fe;"></span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">\theta </span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">y</span></i>}<i><span\; style="color:\; \#2b00fe;">=</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">r</span></i>}\mathrm{<i><span\; style="color:\; \#2b00fe;">sin</span></i>}<i><span\; style="color:\; \#2b00fe;"></span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">\theta </span></i>}$$

For example:

• At ${60}^{\circ }$:$$\mathrm{<i><span\; style="color:\; \#2b00fe;">x</span></i>}<i><span\; style="color:\; \#2b00fe;">=</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">10</span></i>}\mathrm{<i><span\; style="color:\; \#2b00fe;">cos</span></i>}<i><span\; style="color:\; \#2b00fe;"></span></i>{\mathrm{<i><span\; style="color:\; \#2b00fe;">60</span></i>}}^{<i><span\; style="color:\; \#2b00fe;">\circ </span></i>}<i><span\; style="color:\; \#2b00fe;">=</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}<i><span\; style="color:\; \#2b00fe;">,</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">y</span></i>}<i><span\; style="color:\; \#2b00fe;">=</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">10</span></i>}\mathrm{<i><span\; style="color:\; \#2b00fe;">sin</span></i>}<i><span\; style="color:\; \#2b00fe;"></span></i>{\mathrm{<i><span\; style="color:\; \#2b00fe;">60</span></i>}}^{<i><span\; style="color:\; \#2b00fe;">\circ </span></i>}<i><span\; style="color:\; \#2b00fe;">=</span></i>\mathrm{<i><span\; style="color:\; \#2b00fe;">5</span></i>}\sqrt{\mathrm{<i><span\; style="color:\; \#2b00fe;">3</span></i>}}$$

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