Draw on the cardboard as accurately as possible the foregoing diagrams and cut them out. On the interior lines cut the cardboard about half through its thickness. The parts will then readily bend about the half-cut lines into the required form, and can be retained in place by gluing over the edges a strip of paper or linen.

245. PROBLEM. To construct a regular tetraedron.

E

B

At E, the middle point of ABC, erect a perpendicular ED, and take the point D so that AD, which is equal to DB and to DC, shall be equal to AB.

248. Fig. 1 and Fig. 2, following, represent the regular dodecaedron: completed in Fig. 2.

249. Fig. 1, Fig. 2, and Fig. 3, following, represent the regular icosaedron: completed in Fig. 3.

SECTION VI.

MENSURATION OF SOLIDS.

250. In this section it is designed to call special attention, by means of illustrative examples, to all the important rules for finding volumes and surfaces of solids, demonstrated in the preceding sections. Also, methods will be deduced for finding the volumes of certain additional solids, such as the Regular Polyedron, the Wedge, and the Prismoid.

PRISMS AND CYLINDERS.

251. A piece of timber in the form of a right prism is 10 feet long, and contains 48 cubic feet. If a carpenter cuts off one linear foot, how many cubic feet will remain ? See 71, Prop. XIII.

252. A block of marble in the form of a cuboid is 3 feet 6 inches long, 2 feet 3 inches wide, 6 inches thick, and weighs 800 lbs. What is the weight of a like piece of marble whose dimensions are 2 feet 6 inches, by 1 foot 3 inches, by 3 inches? See 73, Prop. XIV.

253. How many cubic feet in each piece of marble described in the foregoing Problem? See 74, Cor. 1.

254. Across a street 50 feet wide a ditch 2 feet broad and 3 feet deep is dug. The ditch is straight, but runs at an angle of 45 degrees with the direction of the street. How many cubic feet of earth were removed in making it? See 79, Cor. 3.

255. inches.

Each edge of the base of a triangular prism is 16 Now, if the altitude is also 16 inches, what is the volume ? See 82.

256. Each edge of the base of a hexagonal prism is 12 inches. Now, if the altitude of this prism is 12 inches, what is its volume? See 90.

257. A cylindrical column is 100 feet high and measures 31.416 feet in circumference. What is its volume ? See 91 and 93.

258. From the column described in Problem 257, 10 linear feet were removed. How many cubic feet remained? See 92.

259. What is the total surface of the cylindrical column described in Problem 257 ? See 96.

260. A cylindrical column is 100 feet high and measures 62.832 feet in circumference. Compare its total surface with that of the column mentioned in Problem 259. See 97. 261. Compare the volumes of the two columns in question. See 92.

The corresponding dimensions of two similar cylinders of revolution are in the ratio of 2 to 3. If the volume of the larger cylinder is 24 cubic feet, what is the volume of the smaller? See 98.

263. Compare the surfaces, lateral and total, of the cylinders described in Problem 262. See 99.

PYRAMIDS AND CONES.

264. Each edge of the base of a right triangular pyramid is 24 feet. Now, if the altitude is also 24 feet, what is

the volume? See 137.

265. The altitude of a right circular cone is 12 feet, and the circumference of its base is 24 feet. What is the volume? See 138.