Surface Areas (A Comparison)

This is the final area of the worksheet on page two, the surface areas, which is column F. Remember we’ve assumed that the side of the cube is one and that means that the surface area of the cube is six square units. The next one to consider is the tetrahedron. There’s the formula and I’ll give you 30 seconds to come up with the answer and of course it’s two root three square units. The next one is the octahedron. There’s the formula. 30 seconds And the answer is root three square units. Next is the icosahedron. Now for this one, you’d better use your calculator and I’ve given you also the value of phi That comes to 1.26 and so on square units. Finally, the dodecahedron there’s the formula, you’ll better use your calculator again and again, you’ll need to find you have five And that’s 7.89 and so on square units. That completes surface areas as far as comparisons go, the three in the middle are interesting. We see that an octahedron inscribed in a tetrahedron is Harford surface area. And a tetrahedron inscribed in a cube has a surface area exactly root three times less than the cube. Because if you multiply two root three by route three, you get six Therefore, the cube is two root three times the surface area of an octahedron, which is in jewel position. I want to demonstrate a slightly different approach to find the area of the dodecahedron. Now, because the surface area of our platonic solids is always made up of regular polygons, we can use a formula that finds the area of a regular polygon and it uses the apathy. There is the apathy. It’s a line, which is at right angles to the perimeter. First, we must find this length. And how are we going to do that? Well, we know the length of each of the sides of our regular pentagon That is they are one over phi the Pentagon is divided into five isosceles triangles. Each has a central angle of 72 degrees. Now the apothem will be that angles bisector and form a top angle of 36 degrees. a right triangle with the base and half the base will be 0.309 We can then use trigonometry. 10 tan is opposite over adjacent. Tan to 36 degrees is 0.309 over the adjacent the height of the epithelium 0.4253 then we can plug that into our formula but before I do that. I recall that the dodecahedron is 12 regular Pentagon’s. So then that will be surface area equals six AP. And then I plug my numbers into that formula. And the answer I get is 7.89 square units. If you recall when we used our formula, which now appears at the top the answer was 7.8 non square units. So, of course, we get the same answer. I wanted to demonstrate how to use the app with them. In certain circumstances, it might be easier. It’s a handy formula to remember when you are dealing with regular polygons. We’ve now completed the worksheet. And that’s really the end of the course. And the next video I just draw a conclusion and tie up a few loose ends.