Math SATs- Geometry

Welcome to a Math SAT Preparation Lesson. In this lesson we will go over simple shape-based geometry.

Ever asked yourself, what in the name of the Great Pickle is math good for? Sure you have. Well, here’s a side of math that’s actually useful for engineers, builders, artists, and anyone else who works with shapes- geometry. You may be surprised to learn that geometry has been around for thousands of years, and ancient Greek and Egyptians are only a few examples of groups that were fascinated by it- just take a look at how exact the shapes of their pyramids and Greek stuff are. And, if any of this stuff doesn’t concern you, you should be aware that geometry will be part of your SAT test, as well as your elementary and high-school math. You should already know a lot of geometry, so this shouldn’t be too hard.

What exactly is geometry?

Geometry deals primarily with sizes, areas, length and volumes of different shapes. One of the shapes we’re concerned about would be the polygon- a planar shape that is bounded and contains a finite number of nodes (vertices). An example of a polygon would be your triangle. Polygons are simple to work with, since they only occupy one 2-dimensional plane.

Here comes a short list of polygons with properties you should know:

1. Triangles- these are polygons with 3 nodes and 3 sides. The sum of the inner angles of a triangle is always 180 degrees. To calculate the perimeter of a triangle, just add up the lengths of the sides. The area of a triangle is always , where b is a base and h is the height (a perpendicular line from that base to the node across).

2. Quadrilaterals- these polygons have 4 sides and 4 nodes. The angles always add up to 360 degrees. There are many different types of these, including the trapezoid (it has a set of 2 parallel opposite sides), the parallelogram (which has 2 sets of 2 parallel opposite sides), the rhombus (all sides are equal), the rectangle (a parallelogram with all right triangles), and the square (a rectangle that’s also a rhombus). The perimeter of a rectangle is , while the area of a rectangle is , and the other shapes are less likely to come up, but their areas can be calculated by dividing them into triangles and rectangles.

3. Anything with more sides and nodes are a bit trickier, but the technique of dividing them into smaller, simpler shapes will always help you figure out how to deal with them. Also, the sum of the angles in a polygon with x sides is always 180(x-2) degrees.

4. Circles- these have no nodes at all, but they’re a shape all right. The area of a perimeter (length of outside line) is given as , with r being the radius, or half the diameter of the circle. The area of a circle is then . If you don’t know this, note that is a button on your calculator that equals about 3.14. Circles also have to do with tangents- lines that touch a circle in just one point. You should know that tangents are always perpendicular to the radius at the point of contact, and trig is usually helpful in situations when you need to use that.

If you think that was easy, good, because now comes the harder part- moving into 3-D.

Geometry isn’t all about planar shapes. There are many more things you’ll have to think about out there. Let’s start:

1. Boxes- these have 6 rectangular sides, with dimensions corresponding to height, length and width. Their surface area is , while their volume is .

2. Cylinders- they have 2 sides that are circles, and a wall put up between them. An example of a cylinder would be what’s left of your toilet paper roll. The volume here would be . The surface area is a bit trickier, being .

3. Prisms- these guys come in many shapes and forms. They’re basically made out of a base that can be any polygon you want, together with rectangular sides. These are tricky, but you can split them up into triangular and rectangular prisms to make your life easier.

An important thing to do when working on a geometry problem- draw it! Sure, the ability to visualize in 3-D will help, but in general drawing stuff out on a piece of paper, and writing down figures for corresponding sides and length will always help you see the question better.

This is a basic summary of the shapes you can expect on your SATs. If you’ve gone through them and think you’re comfortable enough at solving these stuff, try out the geometry test.