Volume of Simple 3D Shapes 06:06 minutes

Video Transcript

Transcript Volume of Simple 3D Shapes

The Mad Hatter is a world-renowned milliner...he makes hats and news of his brilliance has reached the queen. The queen needs a headpiece for a splendid party she's attending. His creations are always the talk of the town but now, he needs boxes for the hats before they're presented to the queen! To do this, he needs to know how to find the volume of simple three dimensional shapes. For the card hat, he needs a rectangular box for the triangular hat, he needs a triangular box, or prism and for the teapot hat, he needs a circular box, or cylinder. The Mad Hatter decides to put his first fashioning in a rectangular box. Let's help him figure out the volume of the boxes he needs. In order to find the volume of any three-dimensional object, you first have to find the area of the base and then multiply the result by the length of the remaining side, which we'll call height. The dimensions of the first hat are: 12 inches for the length 10 inches for the width and 16 inches for the height. Multiplying these two dimensions gives us the area of our first shape, a rectangle. 12 inches times 10 inches is 120 square inches. Then, we multiply 120 by the remaining side, 16, giving us 1,920. Since we're also multiplying the units the units for the volume of the box are ‘inches cubed’. Next, let’s help the Mad Hatter get his Hatter hat into a triangular box. In order to find the volume of the box, which is in the shape of a triangular prism the concept is the same...find the area of one planar face and multiply that by the remaining dimension. The box measurements are: 24 inches for the base of the triangle 16 inches for the height of the triangle and 18 inches for the height of the box. This time, we need to first find the area of the TRIANGLE and then multiply by the remaining dimension, height. We know the formula to find the area of triangles is one-half times the base times the height. Substituting in our measurements and multiplying gives us 192 square inches for the area of the triangular base. Then, multiplying the area of our triangle, 192 square inches, by the remaining dimension, a height of 18 inches gives us the volume of the triangular prism, 3,456 cubic inches. The Mad Hatter is particularly pleased with his last creation, which he’ll put in a special, round box. For this, we only need two measurements: the radius of the hat and its height. The radius of this hat is 25 inches and it’s 51 inches tall. Just like in the other examples, we need to find the area of one planar face and then multiply the result by the remaining dimension. In this case, our planar face is a circle, so we can substitute the equation for the area of a circle in our general equation for volume. Remember that the area of a circle is given by the formula pi ‘r’ squared. We simply have to plug in the values we know to get the area of the circle and then multiply that area by the remaining measurement, height. Doing so gives us 625π square inches for the area. We then need to multiply this by the height, 51 inches, to get the volume. So a box to hold the teapot hat has to have a volume of 31,875π. If we substitute 'pi' with 3.14... We get 100,087.5 cubic inches as the final volume of the cylindrical box. Okay, to review. When finding the volume of simple shapes you first need to find a face of the shape whose area doesn't change with the height. For each of our shapes, we first found the area of the base and then multiplied the result by the height. The area of the rectangular shape is found by multiplying length times width the area of the triangular base is given by the equation one-half base times height and the equation for the area of a circle is pi 'r' squared. Multiplying each of these areas by height gives us volume. Now that the hats are packed away in boxes, let's see which one the queen has chosen. The queen absolutely LOVES the teapot hat but there's something different about THIS hat...