Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Geometry: Part 1

Percentage of Test Level Specifically Assessing Geometry (— = Assumed)

L	E	M	D	A
11%	10%	10%	18%	15%

Geometric concepts are often woven into questions on fractions, percentage, percent change, ratios, and other math and number concepts. TABE word problems often involve plane and solid geometric shapes and your familiarity with a number of concepts: points in a coordinate plane, lines and slopes, angles, polygons, area and perimeter of polygons, the Pythagorean theorem, circles, area and circumference of circles, and volume of solids.

Shapes

Shapes are at the heart of geometry. You will need to know the names of common shapes and their properties. Most of the geometry you will run into is about flat shapes, which leads us to the next section.

Two-Dimensional

Two-dimensional shapes are flat shapes, known in geometry as plane figures. All this means is that every one of these shapes exists totally on a flat surface (a plane). They have two dimensions, a length and a width, but have no thickness rising out of the flat surface.

Comparing Shapes

There are two ways that are useful for comparing shapes. One way is called similarity. Two figures are similar if they have the same shape, but not necessarily the same size. For example, all squares are similar, though some are big and some are small. The two hexagons below are similar. All the angles are the same, but the sides of one aren’t the same lengths as the sides of the other.

The other common way to compare shapes is congruency. Two shapes are congruent if all the angles are the same in each shape, and all the matching sides in each shape have the same length. A good way to think about it is to notice that one figure will fit exactly on top of the other. These two right triangles are congruent.

Shared Attributes

As shown in the list of the most common shapes, many of them are quadrilaterals, meaning having four sides. You should be aware of the ways that some are alike. For example, a rectangle has four sides and four right angles. So does a square. Does that make a square a rectangle? Yes, it does.

Are there quadrilaterals that don’t fit into any of the categories shown? Yes, you could draw any number of them. Here’s one:

Three-Dimensional

As mentioned earlier, plane figures are two-dimensional shapes that have no thickness. On the other hand, there are three-dimensional shapes. These do have a thickness, or you could say they have depth or height, as well as length and width.

Common 3-D Shapes

Comparing Shapes

By their nature, three-dimensional shapes, also called solid shapes, can be more complex than two-dimensional shapes. Some new terms crop up in describing these shapes and we’ll take a look at them below.

Attributes of Shapes

With two-dimensional shapes, we have sides made of line segments. Solid shapes similar to cubes don’t technically have sides, they have flat shapes called faces.

The line segments where the faces meet are called edges.

The point where edges meet is called a vertex, similar to a vertex in a two-dimensional shape. By the way, the plural of vertex is vertices (VER-ti-sees), thanks to the good people who brought us Latin.

The cube above has six faces, twelve edges, and eight vertices.

Other solid shapes like cones, spheres, and cylinders may not have any of these characteristics.

Terms of Comparison

As with two-dimensional shapes, three-dimensional shapes can be congruent if they are the same size and shape. Likewise, they can be similar if they are the same shape, but not the same size.

Working with Shapes

Working with shapes is the business of geometry. In this section we will take a look at dividing, combining, moving, and measuring them.

Partitioning

You should be able to recognize that shapes can be divided, or partitioned, into smaller shapes with equal areas that are fractions of the original shape. For example, a square can be divided into four parts with equal areas.

Each part is \(\frac{1}{4}\) of the original square. The whole square is made of \(\frac{4}{4}\) (four fourths).

Composing

As opposed to dividing shapes, we can put shapes together (compose them) to form new shapes. Below, you can see that we started with a circle and a rectangle. Then the circle was split into two semicircles. The last step was to compose (combine) the three shapes into a new shape.

Manipulating Shapes

Shapes can be moved around in different ways called transformations which are shown below. In each case the original shape is shown in solid lines, and the new transformed shape is drawn in dotted lines.

Picture translation as sliding the original shape with no spin.

Picture rotation as spinning around a point. In this case, the flag is spinning around point O. The point can be on the shape or, as in this case, outside the shape.

Picture reflection as though the shape is being flipped over a line like a pancake. See how the top triangle appears to be pointing down, but the bottom one is pointing up.

Picture dilation as expanding or shrinking while keeping the shape the same.

There’s no reason that you couldn’t do more than one transformation on the same shape, maybe rotate and translate, then dilate.

Measuring Shapes

Two-dimensional shapes have both an area, the amount of flat space they occupy, and a perimeter, the distance around them. With three-dimensional shapes, the area we are often interested in is the area of all surfaces of the shape, called total surface area. Also, every three-dimensional shape has a volume, the total amount of space the shape occupies. Common formulas are shown under the next subheadings.

Perimeter

There are a few formulas for calculating perimeter, but often it’s just a matter of adding the sides of the shape.

Perimeter of a square: \(P=4s\), where \(s\) is the length of a side.

Perimeter of a rectangle: \(P=2l + 2w\), where \(l\) is the length and \(w\) is the width.

Perimeter (circumference) of a circle: \(C= \pi d\), where \(d\) is the diameter. This is an important one to know.

Note: “Pi”, represented by the symbol \(\pi\), is a number used to calculate measurements of circles and is equal to approximately \(3.14\).

Area

To compute area, here are the commonly used formulas for areas of various shapes:

Rectangle: \(A = l \cdot w\), where \(l\) is length and \(w\) is width.

Square: \(A = s^2\), where \(s\) is the length of the side.

Parallelogram: \(A = b \cdot h\), where \(b\) is base and \(h\) is height.

Triangle: \(A = \frac{1}{2}b \cdot h\), where \(h\) is height.

Trapezoid: \(A = \frac{1}{2}(a+b) \cdot h\), where \(a\) is the length of the short side and \(b\) is the length of the long side .

Volume

When flat shapes are raised to a certain height, or depth or thickness, they become solids. These solids occupy space, and that space is called the volume of that solid. Having solved the area, it will be a simple matter of multiplying that area by the thickness to get the volume.

A cube is simply a flat square raised to a thickness equal to the length of the square’s side (\(s\)), therefore the volume (\(V\)) of a cube is:

Cube: \(V = A \cdot s = s^2 \cdot s = s^3\)

Likewise, the volumes of other shapes raised to a thickness or height (\(h\)) become:

Rectangular prism or box with rectangular base: \(V = l \cdot w \cdot h\)

Cylinder: \(V = \pi r^2h\), where \(h\) is the height and \(r\) is the radius.

Cone: \(V = \frac{1}{3} \pi r^2 h\), where \(h\) is the height and \(r\) is the radius.

A sphere is in a category of its own: \(V = \frac{4}{3} \pi r^3\), where \(r\) is the radius.

Surface Area

Three-dimensional shapes can be “unfolded” to create a net. For example, a cube has a top, a bottom, and four sides. If the top, \(T\), and bottom, \(B\), are each cut along three edges, the cube can be opened up to give the figure below.

A net can help with visualizing the total surface area of a solid. It’s pretty clear here that a cube has six faces and each one is a square. If one edge of the cube has a length of \(s\), then each square has a side of \(s\) and an area of \(s^2\). The total surface area of the six faces must then be \(6s^2\).

Another example of a net is below. Picture a cylinder, like a tin can, with its top cut open to make a circle and folded upward. Picture its bottom also cut open and folded downward. Then picture the side of the can cut vertically and unrolled. You would get something like this:

Looking at this net, you can see that a tin can is made of two circles and a rectangle. That can help a lot if you need to find its surface area. It would be \(\pi r^2 + \pi r^2 +lw\).