Question Video: Finding the Area of a Rectangle Given Its Dimensions Mathematics • 8th Grade

Rectangle 𝐴𝐡𝐢𝐷 has 𝐴𝐡 = 25 and 𝐡𝐢 = 36. Draw two perpendiculars segment 𝐡𝐻 and segment 𝐴𝑂 to the plane of the rectangle 𝐴𝐡𝐢𝐷 in the same direction, such that segment 𝐡𝐻 and segment 𝐴𝑂 are both of length 27. What is the area of 𝐢𝐷𝑂𝐻?

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Video Transcript

Rectangle 𝐴𝐡𝐢𝐷 has 𝐴𝐡 equals 25 and 𝐡𝐢 equals 36. Draw two perpendiculars segment 𝐡𝐻 and segment 𝐴𝑂 to the plane of the rectangle 𝐴𝐡𝐢𝐷 in the same direction, such that segment 𝐡𝐻 and segment 𝐴𝑂 are both of length 27. What is the area of 𝐢𝐷𝑂𝐻?

Based on the fact that these line segments are perpendicular to the plane of our rectangle, we know that we’ll be operating in three dimensions. If we let the rectangle 𝐴𝐡𝐢𝐷 be part of this π‘₯𝑦-plane, then the perpendiculars 𝐡𝐻 and 𝐴𝑂 will extend upward in the 𝑧-direction perpendicular to the π‘₯𝑦-plane. Both the perpendiculars measure 27. 𝐴𝐡 measures 25, and 𝐡𝐢 36. We should note here that you could draw many different forms of this diagram. The main purpose of the diagram is to help us visualize these shapes.

Before we calculate the area of 𝐢𝐷𝑂𝐻, let’s see which part of the diagram that would be. 𝐢𝐷𝑂𝐻 is this rectangle on our diagram that I’ve highlighted in pink. To find the area of 𝐢𝐷𝑂𝐻, we need to identify the length and the width. The segment 𝐢𝐷 was part of the original rectangle 𝐴𝐡𝐢𝐷. It’s parallel to line segment 𝐴𝐡 and has the same length, so it’s 25. The width here will be the distance from 𝐢 to 𝐻. Since we know that 𝐡𝐻 is perpendicular to 𝐡𝐢, we can use what we know about right triangles to find the width.

𝐡𝐻 equals 27; 𝐡𝐢 equals 36. We’ll use the Pythagorean theorem, which says 𝑐 squared equals π‘Ž squared plus 𝑏 squared, where 𝑐 is the hypotenuse of a right triangle and π‘Ž and 𝑏 are the other two sides. Our missing side β€” we’ll call 𝑀 squared β€” is equal to 27 squared plus 36 squared, which equals 2025. Taking the square root of both sides, we find that 𝑀 equals 45. If we plug that back into our original diagram, we’ll see that the rectangle 𝐢𝐷𝑂𝐻 has a length of 25 and a width of 45.

To find the area of a rectangle, we multiply the length by the width. When we do that, we find that the area equals 1125. We weren’t given any units, so we can just identify this as units squared. The area of 𝐢𝐷𝑂𝐻 is 1125 units squared.

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