Mathematics Study Guide for the TABE Test

Geometry: Part 3

In this section we will expand on the geometric concepts you’ve learned and explore new ones.

Measuring 2-D and 3-D 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. However, every three-dimensional shape also has volume, the total amount of space the shape occupies. We will discuss the common formulas for these measures below.

Perimeter

There are a few formulas for calculating perimeter, but when there is no simple formula, all you have to do is add the sides of the shape. These are the formulas for the most common shapes:

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

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

• triangle: \(P=a+b+c\), where \(a\), \(b\), and \(c\) are the side lengths.

• circle: \(C= \pi d\), where \(d\) is the diameter and pi (\(\pi\)) is a number used to calculate the measurements of circles, equal to approximately \(3.14\).

Note: The perimeter of a circle is known as a circumference, hence the \(C\) in that formula.

Area

Here are the commonly used formulas for areas of various shapes:

• square: \(A = s^2\), where \(s\) is the length of the sides.

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

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

• rhombus: \(A = p \cdot q\), where \(p\) and \(q\) are the lengths of the diagonals.

• triangle: \(A = \frac{1}{2}b \cdot h\), where \(b\) is the base and \(h\) is the height.

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

• circle: \(A = \pi r^2\), where \(r\) is the radius and \(\pi \approx 3.14\).

Volume

When flat shapes are expanded to a certain depth or thickness, they become three-dimensional shapes or solids. These solids occupy space, and that space is called volume. Determining the volume of a solid simply requires multiplying the area of the original \(2\)-D shape by its new depth.

For instance, 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:

\[V = A \cdot s = s^2 \cdot s = s^3\]

Likewise, the volumes of other shapes raised to a depth or height become:

• rectangular prism: \(V = l \cdot w \cdot h\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height.

• pyramid: \(V=\frac{1}{3}Bh\), where \(B\) is the base and \(h\) is the height.

• cylinder: \(V = \pi r^2h\), where pi (\(\pi\)) \(\approx 3.14\), \(h\) is the height, and \(r\) is the radius.

• cone: \(V = \frac{1}{3} \pi r^2 h\), where \(\pi \approx 3.14\), \(h\) is the height, and \(r\) is the radius.

A sphere is in a category of its own. This is the formula for the volume of a sphere:

\[V = \frac{4}{3} \pi r^3\]

where \(r\) is the radius and \(\pi \approx 3.14\).

Note: Formulas can always be manipulated to fit the data you have. Sometimes you may be given the overall value, such as the area of a rectangle, as well as one of the side lengths, and you will need to determine the other side length.

Surface Area

One measure of three-dimensional shapes is surface area, which is the area of all of its sides. So, if a cube is just a square with depth, the surface area is the area of each of the cube’s six square sides added together.

To help visualize surface area, a \(3\)-D shape can be “unfolded” to create a net. The “net” concept is not on any TABE math test, but the idea is useful for understanding how to find the surface area of a solid shape.

In the next sections, we will explain the surface areas for common \(3\)-D shapes and provide a diagram of the net to help you visualize the calculations.

Rectangular Prism

formula—A rectangular prism describes any \(3\)-D shape that has six rectangular faces. The opposite faces of a rectangular prism are equal in size. The total surface area is given by the formula:

\[2lw + 2lh + 2wh\]

where \(l\) is length, \(w\) is width, and \(h\) is height.

net—A rectangular prism has a top, a bottom, and four sides. If the top and bottom are each cut along three edges, the shape can be opened up to give the figure below (the net):

Cube

formula—A cube is a special type of rectangular prism with six faces that are all squares. 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 cube must then be:

\[6s^2\]

net—A cube unfolds into six squares, all equal in size:

Cylinder

formula—Looking at this shape, you can see that a cylinder (like a tin can) is made of two circles and a rectangle that has been folded into a round shape. You therefore have to add the area of all three to find the surface area. The surface area formula for a cylinder is:

\[\pi r^2 + \pi r^2 +lw\]

net—Picture that 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:

Pyramid

formula—A square pyramid has one square base and four triangular faces. The total surface area is given by the formula:

\[s^2 + 2sl\]

where \(s\) is the base side length and \(l\) is the slant height of each triangle.

net—A square pyramid unfolds into one square base with four triangular faces attached around it:

Cone

formula—A cone has one circular base and one curved surface. The total surface area is given by the formula:

\[\pi r^2 + \pi rl\]

where \(r\) is the radius of the circle and \(l\) is the slant height of the curved surface.

net—A cone unfolds into a circular base and a curved surface shaped like a sector of a circle:

Circles

Circles are a unique type of shape that have no straight edges. In geometry, a circle is defined as the set of all the points that are in the same plane and are the same distance from a center point. For instance, in the circle below, all three named points on the circumference are the same distance from the center point \(O\).

Based on the universal properties of circles, these statements are all true about the circle above:

• The distance from \(O\) to \(B\) is the same as the distance from \(O\) to \(Y\) and \(O\) to \(Z\): \(OB = OY = OZ\).

• Each of the segments \(\overline{OB}\), \(\overline{OY}\), and \(\overline{OZ}\) is a radius (the plural is radii) of the circle.

• The segment \(\overline{BZ}\) goes through the center and is a diameter. A diameter is twice as long as a radius.

The distance around the circle is called the circumference, and it can be calculated using either \(C = \pi d\) or \(C=2\pi r\), where \(d\) is the diameter length, \(r\) is the radius length, and pi (\(\pi\)) is the universal constant approximately equal to \(3.14\).

The circle can be divided into \(360\) degrees, written \(360^\circ\). A half-circle (semicircle) is \(180^\circ\), while a quarter circle is \(90^\circ\), and so on.

Any portion of a circle is called an arc. For example, the part from \(Y\) to \(Z\) is called arc \(YZ\), and its symbol is \(\overset{\frown}{YZ}\).

If you know the number of degrees in an arc, you can calculate its length, To calculate the arc’s length, use this formula:

\[L= \frac{m}{360} \cdot 2 \pi r\]

where \(m\) is the measure of the arc in degrees.

Here’s an example using the figure above.

The arc \(\overset{\frown}{YZ}\) has a measure of \(30^\circ\) and a radius (\(\overline{OY}\)) of \(12\) inches. How long is the arc?

Solution

We simply put the known values in the formula:

\[L= \frac{m}{360} \cdot 2 \pi r\] \[L=\frac{30}{360} \cdot 2 \cdot 3.14 \cdot 12\] \[L= \frac{1}{12} \cdot 75.36\] \[L = 6.28 \text{ in}\]

Using the Coordinate Grid

The coordinate grid is a powerful tool used not just to locate points, but also to draw shapes, measure distances, and solve real-world problems. In this section, you will learn how to use coordinates to create polygons and determine the lengths of their sides.

Before we go into how to use the coordinate grid (or plane), it will be helpful to review the important properties. As you learned earlier, a coordinate grid is formed by two perpendicular number lines:

• the horizontal \(x\)-axis
• the vertical \(y\)-axis

These axes intersect at the origin, \((0,0)\), and divide the plane into four quadrants, as you can see below:

Each point on the grid is written as an ordered pair in this form \((x, y)\):

• The first number (\(x\)) shows how far to move left or right from the origin.
• The second number (\(y\)) shows how far to move up or down.

For example, the point \((3, -2)\) means move three units right and two units down.

Drawing Polygons on the Coordinate Grid

A polygon is a closed figure made up of straight line segments. You can draw polygons on a coordinate grid by plotting points (called vertices) and connecting them in order.

Let’s do an example.

What polygon is created when you graph and connect the points \((1,1)\), \((5,1)\), \((5,4)\), and \((1,4)\) on a coordinate plane?

Solution

First, we will plot each point on the grid. Then, we will connect the points with lines in the given order before connecting the last point back to the first point to close the shape. This is the end result:

This forms a rectangle. Every polygon that you’ve learned about can be graphed on a coordinate grid.

Finding Side Lengths on the Grid

As you learned previously, when two points share the same \(x\)-coordinate or the same \(y\)-coordinate, the distance between them can be found easily by measuring the segment between the points:

• If the \(y\)-coordinates are the same, the segment is horizontal.
• If the \(x\)-coordinates are the same, the segment is vertical.

This fact is useful when finding the side lengths of a polygon that is drawn within the coordinate grid. As you did earlier with basic line segments, you simply take the absolute value of the difference between the coordinates at the vertices of each side.

Once you can find side lengths, you can determine properties of shapes such as perimeter. Let’s look at the rectangle we graphed before and find its perimeter.

• bottom side: \(\vert 5 - 1 \vert = 4\)
• right side: \(\vert 4 - 1 \vert = 3\)
• top side: \(4\) (same as bottom)
• left side: \(3\) (same as right)

So, the perimeter is:

\[4 + 3 + 4 + 3 = 14\]

Coordinate grids are used in many real-world situations, such as maps, construction layouts, and computer graphics. As such, let’s do an example related to real life.

A park is designed using a coordinate grid. The corners of a rectangular playground are at \((2,2)\), \((8,2)\), \((8,6)\), and \((2,6)\). How much fencing is needed to enclose the playground?

Solution

We need to find the perimeter of the playground, which is a rectangle. The perimeter of a rectangle is found using this formula:

\[P = 2(l+w)\]

where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. Let’s find the length and width:

• length: \(\vert{8 - 2}\vert = 6\)
• width: \(\vert{6 - 2}\vert = 4\)

Thus, the perimeter is:

\[P = 2(6+4) = 2(10) = 20\]

Scale Drawings

When you learned about manipulating shapes earlier, you read about the concept of dilation, or making a shape bigger or smaller. Another way of referring to dilation is scaling a shape up or down.

Consider the drawing below with two scale drawings of a wedge. The larger triangle, \(\triangle CAT\), has a scale where once inch equals three feet (represented as \(1’’ = 3'\)), and the smaller triangle, \(\triangle DOG\), has a scale of \(1’’ = 6'\). In other words, the triangles are similar, but \(\triangle CAT\) is scaled up.

Scale drawings are often used to represent a real-world shape. Think of the way a map is used to show the layout of a city or a country. Obviously, a map can’t be the actual size of a city, so it’s a scaled down version of the city. For the map to be accurate, though, it needs to maintain the same proportions of the actual land it represents.

Now consider the map below, where three cities—Seattle, Portland, and Boise—are plotted on a coordinate grid. Their locations are given by the following coordinates:

• Seattle: \((0, 0)\)
• Portland: \((3, 2)\)
• Boise: \((8, 0)\)

By plotting these points and connecting them, we create a polygon in the coordinate plane:

This shows how scale drawings and coordinate geometry can work together to represent real-world locations.

We can use this information to determine information about the real world, such as distance. Suppose the map uses a scale where one unit on the grid represents \(50\) miles.

First, let’s find the distance from Seattle to Boise. These two points lie on the same horizontal line (\(y=0\)), so we only need to compare their \(x\)-coordinates:

\[\vert 8 - 0 \vert = 8\]

So the total distance is eight units. Using the scale, we have:

\[8 \times 50 = 400 \text{ mi}\]

The distance between Seattle and Boise is \(400\) miles (according to our graphic).

Now, let’s estimate the distance from Seattle to Portland. From \((0,0)\) to \((3,2)\), we move three units to the right and two units up. This creates a diagonal distance across the grid, which is a little more than three units. From the graph, we can estimate it to be about \(3.6\) units.

Note: This is an approximate value; in the next section, you’ll learn how to find the exact value.

Therefore, using the scale, we get:

\[3.6 \times 50 \approx 180 \text{ mi}\]

The Pythagorean Theorem

The Pythagorean theorem is a pattern that applies to all right triangles. It is used to determine the side lengths of triangles, such as this one:

In the right triangle above, sides \(a\) and \(b\) are the shorter sides and are called the legs of the triangle. The longest side, \(c\), is called the hypotenuse. The Pythagorean theorem states that the sum of the squares of the two legs equals the square of the hypotenuse. In terms of \(a, b,\) and \(c\), the theorem is written as such:

\[a^2 + b^2 = c^2\]

In this formula, \(a\) and \(b\) are always the legs of the right triangle, and \(c\) is always the hypotenuse. This lets us figure out the length of any side if we know the other two sides. Here’s an example:

How long is side \(x\) in this triangle?

We can let \(a=6\), \(b=8\), and \(c=x\).

So:

\[6^2 + 8^2 = x^2\] \[36 + 64 = x^2\] \[100 = x^2\] \[\sqrt{100} = \sqrt{x^2}\] \[10 = x\]

In the real world, the Pythagorean theorem is used all the time. For instance, when building a house, carpenters could measure the diagonals of the foundation and use the Pythagorean theorem to make sure all the corners are right angles.

Suppose the drawing below represents the layout of a small tool shed. Given the dimensions shown, will the shed be a true rectangle? Remember, you can’t go by looks, only by the given numbers.

We just need to see if the dimensions are in line with the Pythagorean theorem.

Is \(9^2 + 12^2 = 14^2\) true? We’ll do the math and figure it out:

\[81 + 144 = 196\] \[225 \neq 196\]

It is not true, so these dimensions don’t make a true rectangle.

The Pythagorean theorem can also be used in the coordinate plane.

Suppose one point is at \((2, 5)\) and another point is at \((5, 1)\). How far apart are the points?

If you sketch a segment from \(A\) to \(B\), you can see that we can draw a right triangle with \(\overline {AB}\) as a hypotenuse and legs with lengths \(3\) and \(4\). The Pythagorean theorem tells us that \(a^2+b^2=c^2\), so let \(a=3\), \(b=4\), and \(c=\overline{AB}\). Now, you can calculate the length of \(c\):

\[3^2+4^2=c^2\] \[9+16=c^2\] \[25 = c^2\] \[\sqrt{25} = \sqrt{c^2}\] \[5 = c\]

Which means that segment \(\overline{AB}\) equals \(5\).