Chapter 6 Solid Geometry Questions about solid geometry frequently test plane geometry techniques. They’re dicult mostly because the added third dimension makes them harder to visualize. You’re likely to run into three or four solid geometry questions on either one of the Math Subject Tests, however, so it’s important to practice. If you’re not the artistic type and have trouble drawing cubes, cylinders, and so on, it’s worthwhile to practice sketching the shapes in the following pages. The ability to make your own drawing is often helpful. PRISMS Prisms are three-dimensional gures that have two parallel bases that are polygons. Cubes and rectangular solids are examples of prisms that ETS often asks about. In general, the volume of a prism is given by the following formula: Volume of a Prism V = Bh In this formula, B represents the area of either base of the prism (the top or the bottom), and h represents the height of the prism (perpendicular to the base). The formulas for the volume of a rectangular solid, a cube, and a cylinder all come from this basic formula. Area and Volume In general the volume of a shape involves the area of the base, often referred to as B, and the height, or h, of the solid. RECTANGULAR SOLID A rectangular solid is simply a box; ETS also sometimes calls it a rectangular prism. It has three distinct dimensions: length, width, and height. The volume of a rectangular solid (the amount of space it contains) is given by this formula: Volume of a Rectangular Solid V = lwh The surface area (SA) of a rectangular solid is the sum of the areas of all of its faces. A rectangular solid’s surface area is given by the formula on the next page. Surface Area of a Rectangular Solid SA = 2lw + 2wh + 2lh The volume and surface area of a solid make up the most basic information you can have about that solid (volume is tested more often than surface area). You may also be asked about lengths within a rectangular solid—edges and diagonals. The dimensions of the solid give the lengths of its edges, and the diagonal of any face of a rectangular solid can be found using the Pythagorean theorem. There’s one more length you may be asked about—the long diagonal (or space diagonal) that passes from corner to corner through the center of the box. The length of the long diagonal is given by this formula: Long Diagonal of a Rectangular Solid (Super Pythagorean Theorem) a2 + b2 + c2 = d2 This is the Pythagorean theorem with a third dimension added, and it works just the same way. This formula will work in any rectangular box. The long diagonal is the longest straight line that can be drawn inside any rectangular solid. CUBES A cube is a rectangular solid that has the same length in all three dimensions. All six of its faces are squares. This simplies the formulas for volume, surface area, and the long diagonal. Volume of a Cube V = s3 Surface Area of a Cube SA = 6s2 Face Diagonal of a Cube f = s Long Diagonal of a Cube d = s CYLINDERS A cylinder is like a prism but with a circular base. It has two important dimensions—radius and height. Remember that volume is the area of the base times the height. In this case, the base is a circle. The area of a circle is πr2. So the volume of a cylinder is πr2h. Volume of a Cylinder V = πr2h The surface area of a cylinder is found by adding the areas of the two circular bases to the area of the rectangle you’d get if you unrolled the side of the cylinder. That boils down to the following formula: Surface Area of a Cylinder SA = 2πr2 + 2πrh The longest line that can be drawn inside a cylinder is the diagonal of the rectangle formed by the diameter and the height of the cylinder. You can find its length with the Pythagorean theorem. d2 = (2r)2 + h 2 CONES If you take a cylinder and shrink one of its circular bases down to a point, then you have a cone. A cone has three signicant dimensions which form a right triangle—its radius, its height, and its slant height, which is the straight-line distance from the tip of the cone to a point on the edge of its base. The formulas for the volume and surface area of a cone are given in the information box at the beginning of both of the Math Subject Tests. The formula for the volume of a cone is pretty straightforward: Volume of a Cone V = πr2h Connect the Dots Notice that the volume of a cone is just one-third of the volume of a circular cylinder. Make memorizing easy! But you have to be careful computing surface area for a cone using the formula provided by ETS. The formula at the beginning of the Math Subject Tests is for the lateral area of a cone—the area of the sloping sides—not the complete surface area. It doesn’t include the circular base. Here’s a more useful equation for the surface area of a cone. Surface Area of a Cone SA = πrl + πr2 If you want to calculate only the lateral area of a cone, just use the first half of the above formula—leave the πr2 off. SPHERES A sphere is simply a hollow ball. It can be dened as all of the points in space at a xed distance from a central point. The one important measure in a sphere is its radius. The formulas for the volume and surface area of a sphere are given to you at the very beginning of both Math Subject Tests. That means that you don’t need to have them memorized, but here they are anyway: Volume of a Sphere V = πr3 Surface Area of a Sphere SA = 4πr2 The intersection of a plane and a sphere always forms a circle unless the plane is tangent to the sphere, in which case the plane and sphere touch at only one point. PYRAMIDS A pyramid is a little like a cone, except that its base is a polygon instead of a circle. Pyramids don’t show up often on the Math Subject Tests. When you do run into a pyramid, it will almost always have a rectangular base. Pyramids can be pretty complicated solids, but for the purposes of the Math Subject Tests, a pyramid has just two important measures—the area of its base and its height. The height of a pyramid is the length of a line drawn straight down from the pyramid’s tip to its base. The height is perpendicular to the base. The volume of a pyramid is given by this formula. Connect the Dots Notice that the volume of a pyramid is just one-third of the volume of a prism. Make memorizing easy! Volume of a Pyramid V = Bh (B = area of base) It’s not really possible to give a general formula for the surface area of a pyramid because there are so many dierent kinds. At any rate, the information is not generally tested on the Math Subject Tests. If you should be called upon to gure out the surface area of a pyramid, just gure out the area of each face using polygon rules, and add them up. TRICKS OF THE TRADE Here are some of the most common solid geometry question types you’re likely to encounter on the Math Subject Tests. They occur much more often on the Math Level 2 Subject Test than on the Math Level 1 Subject Test, but they can appear on either test. Triangles in Rectangular Solids Many questions about rectangular solids are actually testing triangle rules. Such questions generally ask for the lengths of the diagonals of a box’s faces, the long diagonal of a box, or other lengths. These questions are usually solved using the Pythagorean theorem and the Super Pythagorean theorem that nds a box’s long diagonal (see the section on Rectangular Solids). DRILL Here are some practice questions using triangle rules in rectangular solids. The answers to these drills can be found in Chapter 12. 32. What is the length of the longest line that can be drawn in a cube of volume 27 ? (A) 3.0 (B) 4.2 (C) 4.9 (D) 5.2 (E) 9.0 36. In the rectangular solid shown, if AB = 4, BC = 3, and BF = 12, what is the perimeter of triangle EDB ? (A) 27.33 (B) 28.40 (C) 29.20 (D) 29.50 (E) 30.37 39. In the cube above, M is the midpoint of BC, and N is the midpoint of GH. If the cube has a volume of 1, what is the length of MN ? (A) 1.23 (B) 1.36 (C) 1.41 (D) 1.73 (E) 1.89 Volume Questions Many solid geometry questions test your understanding of the relationship between a solid’s volume and its other dimensions— sometimes including the solid’s surface area. To solve these questions, just plug the numbers you’re given into the solid’s volume formula. Drill Try the following practice questions. The answers to these drills can be found in Chapter 12. 17. The volume and surface area of a cube are equal. What is the length of an edge of this cube? (A) 1 (B) 2 (C) 4 (D) 6 (E) 9 24.