8.5 SAT Math - Solid Shapes, Shaded Regions, and Perception

The solid shapes on the SAT math section are not usually that difficult.  Most of them deal with either volume or surface area.

Volume

Volume is the interior of a three-dimensional object.  The volume of most solids can be found by multiplying the area of the base times the height, so the volume of a rectangular solid is l•w•h, the volume of a cube is s³, and the volume of a cylinder is πr²h.

A rectangular solid is one kind of prism.  You may occasionally run into another shape of prism such as a triangular or hexagonal prism.  A prism is a solid in which the top and bottom are a shape (like a triangle), and the sides are rectangles.  All prisms have the same formula for volume: area of the base times the height.

Surface Area

The surface area of a solid is found by adding up the areas of all the faces.
-SAT Math Hint: Don't forget about the ones on the back that you can't see in the drawing!  Rectangular solids and cubes have 6 faces.

To find the surface area of a rectangular solid, multiply the l times the w, the w by the h, and the l times the h.  Multiply each one of those answers by 2 to account for the back sides, and then add all those numbers together.

The surface area of a cube is much faster since all the sides are the same.  To find surface area of a cube, simply use the formula 6s2.

A cylinder's surface area is a little trickier, but luckily, is rarely tested.  If you do come across a surface area of a cylinder question, just remember that the surface area is the sides – so think of a soup can label.  If you peel off the label, what shape do you have?  A rectangle.  The width of that rectangle is the cylinder's height, but what's the length?  The length of that rectangle is the circumference of the circular base – that's the distance around the cylinder.  From there, you'll also need to add the top and bottom of the cylinder.  So the entire formula for surface area of a cylinder is 2πrh + 2πr².

Another rare question you might get deals with the diagonal of a rectangular solid.  The diagonal is found by drawing a line from one corner to the exact opposite corner, like this:

The formula for finding the length of the diagonal is d^{2 =} l² + w² + h² .

Cones, Pyramids and Spheres

Cones, Pyramids and Spheres don't show up that often on this test, and when they do they most likely won't be testing you on volume.  You may be tested on surface area, or more often, ratios of the top to the bottom or something like that.  If they are testing you on volume or anything more obscure, they will give you the formula in the body of the SAT question.

A cone looks like this:

where r is the radius, h is the height, and l is the slant height.
            -SAT Math Hint: Don't confuse the slant height with the height.  The height is used to find volume.  The slant height is used to find surface area.

A pyramid looks like this:

The base of a pyramid is a rectangle.

In both cones and pyramids, if you draw a line parallel to the base that divides a top portion from the lower portion, the smaller cone or pyramid formed by that line will be similar to the original cone or pyramid.  This is most often what will be tested.

A sphere looks like this:

Like a circle, all radii in a sphere are equal.-+