Solid As A Rock
Solid As A Rock
Solids are three-dimensional shapes, with the dimension of depth added to length and height. With solids, there’s good news and bad news. The bad news is that solids can be difficult to visualize. But the good news more than makes up for it: The only solids on the new SAT are cubes, rectangular solids, and right cylinders. Learn to visualize these three shapes now, before the test, and you’ll be fine.
Rectangular Solids
A rectangular solid is a prism with a rectangular base and edges that are perpendicular to its base. It looks a lot like a cardboard box.
A rectangular solid has three important dimensions: length (l), width (w), and height (h). If you know these three measurements, you can find the solid’s surface area, volume, and diagonal length.
Volume of a Rectangular Solid
The formula for the volume of a rectangular solid takes the formula for area of a rectangle and adds another dimension. The area of a rectangle is A = lh (area equals length times height). The formula for the volume of a rectangular solid adds on width:
Here’s a good old-fashioned example:
What is the volume of the figure presented below?
Just plug the values into the volume formula and you’re good to go: V = (3x)(2x)(x) = 6x3.
Surface Area of a Rectangular Solid
The surface area of a solid is the area of its outermost skin. In the case of rectangular solids, imagine a cardboard box all closed up. The surface of that closed box is made of six rectangles: the sum of the areas of the six rectangles is the surface area of the box. To make things even easier, the six rectangles come in three congruent pairs. We’ve marked the congruent pairs by shades of gray in the image below: One pair is clear, one pair is light gray, and one pair is dark gray.
Two faces have areas of l w, two faces have areas of l h, and two faces have areas of w h. The surface area of the entire solid is the sum of the areas of the congruent pairs:
Wanna practice? Alright. What’s the surface area of this guy?
Plug in. Plug in. Plug in.
Meat Cleaver Problems
The SAT won’t just hand you surface area questions on a silver platter. It’ll make you work for them. One of the ways the SAT likes to make you work goes like this. A question will describe a solid, give you all of its measurements, and then tell you that the box has been cut in half. You’ll then have to find the combined surface area of the two new boxes. For example, pictured below is a rectangular solid that has a length of 8, a depth of 4, and a height of 4. Then, out of the blue, a giant cleaver comes down and cuts the solid into two cubes.
As you can see from the diagram, when the rectangle was cut in two, two new surfaces suddenly appeared (these are the darkened surfaces). But notice that the total volume of the two cubes has remained the same. So here’s a rule: Whenever a solid is cut into smaller pieces, its surface area increases, but its volume is unchanged. The SAT loves to test this little factoid.
Diagonal Length of a Rectangular Solid
The diagonal of a rectangular solid, d, is the line segment whose endpoints are opposite corners of the solid. Every rectangular solid has four diagonals, each with the same length, that connect each pair of opposite vertices. Here’s one diagonal drawn in:
It’s possible that an SAT question will test to see if you can find the length of a diagonal. Now you can:
where l is the length, w is the width, and h is the height. The formula is like a pumped up Vin Diesel version of the Pythagorean theorem. Check it out in action:
What is the length of diagonal AH in the rectangular solid below if AC = 5, GH = 6, and CG = 3?
The question gives the length, width, and height of the rectangular solid, so you can just plug those numbers into the formula:
A cube is a square brought into 3-D. The length, width, and height of a cube are equal, and each of its six faces is a square.
Volume of a Cube
The formula for finding the volume of a cube is essentially the same as the formula for the volume of a rectangular volume. However, since a cube’s length, width, and height are all equal, the formula for the volume of a cube is
where s is the length of one edge of the cube.
Surface Area of a Cube
Since a cube is just a rectangular solid whose sides are all equal, the formula for finding the surface area of a cube is the same as the formula for finding the surface area of a rectangular solid, except with s substituted in for l, w, and h:
Diagonal Length of a Cube
The formula for the diagonal of a cube is also adapted from the formula for the diagonal length of a rectangular solid, with s substituted for l, w, and h.
Right Circular Cylinders
A right circular cylinder looks like one of those cardboard things that toilet paper comes on, except it isn’t hollow. In fact, one way to think of a right circular cylinder is as a rectangle curved around so that its ends meet.
A right circular cylinder has two connected congruent circular bases and looks like this:
The height of a cylinder, h, is the length of the line segment whose endpoints are the centers of the bases. The radius of a cylinder, r, is the radius of its base. For the new SAT, all you need to know about a right circular cylinder is how to calculate its volume.
Volume of a Cylinder
The volume of a cylinder is the product of the area of its base and its height. Because a cylinder has a circular base, the volume of a cylinder is equal to the area of the circle that is the base times the height:
Try to find the volume of the cylinder below:
This cylinder has a radius of 4 and a height of 6. Using the volume formula,
Sketchy Word Problems
The SAT has been known to ask word problems about solids. Because solids are so difficult to visualize, these problems can seem brutally difficult. So here’s the rule: Always sketch out what the question is describing. Once you see what the question’s talking about, you seldom have to do much more than plug the right numbers into the right equation.
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