Video: Finding the Total Surface Area and the Volume of a Cuboid

Using the figures, determine the ratio between the area of triangle 𝐴𝐵𝐶 and the area of square 𝑋𝑌𝑍𝐿. Give your answer in its simplest form.

02:25

Video Transcript

Using the figures, determine the ratio between the area of triangle 𝐴𝐵𝐶 and the area of square 𝑋𝑌𝑍𝐿. Give your answer in its simplest form.

We’re given two shapes, and we want to find the ratio of the area of the triangle to the area of the square. In order to do that, we’ll need to know the area of the triangle and the area of the square. We find the area of a triangle as one-half times the height times the base and the area of a square is its side squared.

Starting with the triangle, we have a height of 20 and a base of 32. It’s worth checking here and noting that we are dealing with the same units for the height and the base. One-half times 20 is 10, and 10 times 32 is 320. The units are centimeters squared, and the area of this triangle is 320 centimeters squared. We also want to check and make sure that the units we’re using for the square are the same as for the triangle. And they are; they’re in centimeters. The area of the square will be 20 squared, which is 400. And again, our units are centimeters squared.

To put this ratio together, we would have 320 to 400. But, again, we want the simplest form. And that means we can divide both of these values by 10, which gives us 32 to 40. But 32 and 40 can be simplified even further since they’re both divisible by eight. 32 divided by eight is four, and 40 divided by eight is five. And this gives us the ratio of the area of the triangle to the area of the square in simplest form, four to five.

Before we move on, we should also note a common misconception. If you took the base of the triangle, 32, and made a ratio of the square base, you would have 32 to 20. And when you simplify that, you get eight to five. And while it is true that this is the ratio from the triangle’s base to the square base, it is not true that this would then be the ratio for the areas. In order to calculate the ratio of the areas, you must find the areas of both of these figures, which would give you four to five.

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