Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

Part IV. Picturing and Measuring - Graphs, Measures, Stats, and Sets

Chapter 16. Picture This: Basic Geometry

In This Chapter

Knowing the basic components of geometry: points, lines, angles, and shapes

Examining two-dimensional shapes

Looking at solid geometry

Finding out how to measure a variety of shapes

G eometry is the mathematics of figures such as squares, circles, triangles, and lines. Because geometry is the math of physical space, it's one of the most useful areas of math. Geometry comes into play when measuring rooms or walls in your house, the area of a circular garden, the volume of water in a pool, or the shortest distance across a rectangular field.

Although geometry is usually a yearlong course in high school, you may be surprised by how quickly you can pick up what you need to know about basic geometry. Much of what you discover in a geometry course is how to write geometric proofs, which you don't need for algebra — or trigonometry, or even calculus.

In this chapter, I give you a quick and practical overview of geometry. First, I show you four important concepts in plane geometry: points, lines, angles, and shapes. Then I give you the basics on geometric shapes, from flat circles to solid cubes. Finally, I discuss how to measure geometric shapes by finding the area and perimeter of two-dimensional forms and the volume and surface area of some geometric solids.

Of course, if you want to know more about geometry, the ideal place to look beyond this chapter is Geometry For Dummies, 2nd Edition, by Mark Ryan (published by Wiley)!

Getting on the Plane: Points, Lines, Angles, and Shapes

Plane geometry is the study of figures on a two-dimensional surface — that is, on a plane. You can think of the plane as a piece of paper with no thickness at all. Technically, a plane doesn't end at the edge of the paper — it continues forever.

In this section, I introduce you to four important concepts in plane geometry: points, lines, angles, and shapes (squares, circles, triangles, and so forth).

Making some points

A point is a location on a plane. It has no size or shape. Although in reality a point is too small to be seen, you can represent it visually in a drawing by using a dot.

Illustration by Wiley, Composition Services Graphics

When two lines intersect, as shown in this figure, they share a single point. Additionally, each corner of a polygon is a point. (Keep reading for more on lines and polygons.)

Knowing your lines

A line — also called a straight line — is pretty much what it sounds like; it marks the shortest distance between two points, but it extends infinitely in both directions. It has length but no width, making it a one-dimensional (1-D) figure.

Given any two points, you can draw exactly one line that passes through both of them. In other words, two points determine a line.

Illustration by Wiley, Composition Services Graphics

When two lines intersect, they share a single point. When two lines don't intersect, they are parallel, which means that they remain the same distance from each other everywhere. A good visual aid for parallel lines is a set of railroad tracks. In geometry, you draw a line with arrows at both ends. Arrows on either end of a line mean that the line goes on forever (as you can see in Chapter 1, where I discuss the number line).

A line segment is a piece of a line that has endpoints, as shown here.

Illustration by Wiley, Composition Services Graphics

A ray is a piece of a line that starts at a point and extends infinitely in one direction, kind of like a laser. It has one endpoint and one arrow.

Illustration by Wiley, Composition Services Graphics

Figuring the angles

An angle is formed when two rays extend from the same point.

Illustration by Wiley, Composition Services Graphics

Angles are typically used in carpentry to measure the corners of objects. They're also used in navigation to indicate a sudden change in direction. For example, when you're driving, it's common to distinguish when the angle of a turn is “sharp” or “not so sharp.”

The sharpness of an angle is usually measured in degrees. The most common angle is the right angle — the angle at the corner of a square — which is a 90° (90-degree) angle:

Illustration by Wiley, Composition Services Graphics

Angles that have fewer than 90° — that is, angles that are sharper than a right angle — are called acute angles, like this one:

Illustration by Wiley, Composition Services Graphics

Angles that measure greater than 90° — that is, angles that aren't as sharp as a right angle — are called obtuse angles, as seen here:

Illustration by Wiley, Composition Services Graphics

When an angle is exactly 180°, it forms a straight line and is called a straight angle.

Illustration by Wiley, Composition Services Graphics

Shaping things up

A shape is any closed geometrical figure that has an inside and an outside. Circles, squares, triangles, and larger polygons are all examples of shapes.

Much of plane geometry focuses on different types of shapes. In the next section, I show you how to identify a variety of shapes. Later in this chapter, I show you how to measure these shapes.

Closed Encounters: Shaping Up Your Understanding of 2-D Shapes

A shape is any closed two-dimensional (2-D) geometrical figure that has an inside and an outside, separated by the perimeter (boundary) of the shape. The area of a shape is the measurement of the size inside that shape.

A few shapes that you're probably familiar with include the square, rectangle, and triangle. However, many shapes don't have names, as you can see in Figure 16-1.

Illustration by Wiley, Composition Services Graphics

Figure 16-1: Unnamed shapes.

Measuring the perimeter and area of shapes is useful for a variety of applications, from land surveying (to get information about a parcel of land that you're measuring) to sewing (to figure out how much material you need for a project). In this section, I introduce you to a variety of geometric shapes. Later in the chapter, I show you how to find the perimeter and area of each, but for now, I just acquaint you with them.

Polygons

A polygon is any shape whose sides are all straight. Every polygon has three or more sides (if it had fewer than three, it wouldn't really be a shape at all). Following are a few of the most common polygons.

Triangles

The most basic shape with straight sides is the triangle, a three-sided polygon. You find out all about triangles when you study trigonometry (and what better place to begin than Trigonometry For Dummies, 2nd Edition, by Mary Jane Sterling [Wiley]?). Triangles are classified on the basis of their sides and angles. Take a look at the differences (and see Figure 16-2):

· Equilateral: An equilateral triangle has three sides that are all the same length and three angles that all measure 60°.

· Isosceles: An isosceles triangle has two sides that are the same length and two equal angles.

· Scalene: Scalene triangles have three sides that are all different lengths and three angles that are all unequal.

· Right: A right triangle has one right angle. It may be isosceles or scalene.

Illustration by Wiley, Composition Services Graphics

Figure 16-2: Types of triangles.

Quadrilaterals

A quadrilateral is any shape that has four straight sides. Quadrilaterals are one of the most common shapes you see in daily life. If you doubt this statement, look around and notice that most rooms, doors, windows, and tabletops are quadrilaterals. Here I introduce you to a few common quadrilaterals (Figure 16-3 shows you what they look like):

· Square: A square has four right angles and four sides of equal length; also, both pairs of opposite sides (sides directly across from each other) are parallel.

· Rectangle: Like a square, a rectangle has four right angles and two pairs of opposite sides that are parallel. Unlike the square, however, although opposite sides are equal in length, sides that share a corner — adjacent sides — may have different lengths.

· Rhombus: Imagine starting with a square and collapsing it as if its corners were hinges. This shape is called a rhombus. All four sides are equal in length, and both pairs of opposite sides are parallel.

· Parallelogram: Imagine starting with a rectangle and collapsing it as if the corners were hinges. This shape is a parallelogram — both pairs of opposite sides are equal in length, and both pairs of opposite sides are parallel.

· Trapezoid: The trapezoid's only important feature is that at least two opposite sides are parallel.

· Kite: A kite is a quadrilateral with two pairs of adjacent sides that are the same length.

Illustration by Wiley, Composition Services Graphics

Figure 16-3: Common quadrilaterals.

A quadrilateral can fit into more than one of these categories. For example, every parallelogram (with two sets of parallel sides) is also a trapezoid (with at least one set of parallel sides). Every rectangle and rhombus is also both a parallelogram and a trapezoid. And every square is also all five other types of quadrilaterals. In practice, however, it's common to identify a quadrilateral as descriptively as possible — that is, use the first word in the list that accurately describes it.

Polygons on steroids — larger polygons

A polygon can have any number of sides. Polygons with more than four sides aren't as common as triangles and quadrilaterals, but they're still worth knowing about. Larger polygons come in two basic varieties: regular and irregular.

A regular polygon has equal sides and equal angles. The most common are regular pentagons (five sides), regular hexagons (six sides), and regular octagons (eight sides). See Figure 16-4.

Illustration by Wiley, Composition Services Graphics

Figure 16-4: A pentagon, a hexagon, and an octagon.

Every other polygon is an irregular polygon (see Figure 16-5).

Illustration by Wiley, Composition Services Graphics

Figure 16-5: Various irregular polygons.

Circles

A circle is the set of all points that are a constant distance from the circle's center. The distance from any point on the circle to its center is called the radius of the circle. The distance from any point on the circle straight through the center to the other side of the circle is called the diameter of the circle.

Unlike polygons, a circle has no straight edges. The ancient Greeks — who invented much of geometry as we know it today — thought that the circle was the most perfect geometric shape.

Taking a Trip to Another Dimension: Solid Geometry

Solid geometry is the study of shapes in space — that is, the study of shapes in three dimensions. A solid is the spatial (three-dimensional, or 3-D) equivalent of a shape. Every solid has an inside and an outside separated by the surface of the solid. Here, I introduce you to a variety of solids.

The many faces of polyhedrons

A polyhedron is the three-dimensional equivalent of a polygon. As you may recall from earlier in the chapter, a polygon is a shape that has only straight sides. Similarly, a polyhedron is a solid that has only straight edges and flat faces (that is, faces that are polygons).

The most common polyhedron is the cube (see Figure 16-6). As you can see, a cube has 6 flat faces that are polygons — in this case, all the faces are square — and 12 straight edges. Additionally, a cube has eight vertexes, or vertices (corners). Later in this chapter, I show you how to measure the surface area and volume of a cube.

Illustration by Wiley, Composition Services Graphics

Figure 16-6: A typical cube.

Figure 16-7 shows a few common polyhedrons (or polyhedra).

Illustration by Wiley, Composition Services Graphics

Figure 16-7: Common polyhedrons.

Later in this chapter, I show you how measure each of these polyhedrons to determine its volume — that is, the amount of space contained inside its surface.

One special set of polyhedrons is called the five regular solids (see Figure 16-8). Each regular solid has identical faces that are regular polygons. Notice that a cube is a type of regular solid. Similarly, the tetrahedron is a pyramid with four faces that are equilateral triangles.

Illustration by Wiley, Composition Services Graphics

Figure 16-8: The five regular solids.

3-D shapes with curves

Many solids aren't polyhedrons because they contain at least one curved surface. Here are a few of the most common of these types of solids (also see Figure 16-9):

· Sphere: A sphere is the solid, or three-dimensional, equivalent of a circle. A ball is a perfect visual aid for a sphere.

· Cylinder: A cylinder has a circular base and extends vertically from the plane. A good visual aid for a cylinder is a can of soup.

· Cone: A cone is a solid with a round base that extends vertically to a single point. A good visual aid for a cone is an ice-cream cone.

Illustration by Wiley, Composition Services Graphics

Figure 16-9: Spheres, cylinders, and cones.

In the next section, I show you how to measure a sphere and a cylinder to determine their volume — that is, the amount of space contained within.

Measuring Shapes: Perimeter, Area, Surface Area, and Volume

In this section, I introduce you to some important formulas for measuring shapes on the plane and solids in space. These formulas use letters to stand for numbers that you can plug in to make specific measurements. Using letters in place of numbers is a feature you'll see more of in Part V, when I discuss algebra.

2-D: Measuring on the flat

Two important skills in geometry — and real life — are finding the perimeter and calculating the area of shapes. A shape's perimeter is a measurement of the length of its sides. You use perimeter for measuring the distance around the edges of a room, building, or circular pathway. A shape's area is a measurement of how big it is inside. You use area when measuring the size of a wall, a table, or a tire.

For example, in Figure 16-10, I give you the lengths of the sides of each shape.

Illustration by Wiley, Composition Services Graphics

Figure 16-10: Measuring the sides of figures.

When every side of a shape is straight, you can measure its perimeter by adding up the lengths of all its sides.

Similarly, in Figure 16-11, I give you the area of each shape.

Illustration by Wiley, Composition Services Graphics

Figure 16-11: The areas of figures.

The area of a shape is always measured in square units: square inches (in.²), square feet (ft.²), square miles (mi.²), square kilometers (km²), and so on — even if you're talking about the area of a circle! (For more on measurements, flip to Chapter 15.)

I cover these types of calculations in this section. (For more information on the names of shapes, refer to “Closed Encounters: Shaping Up Your Understanding of 2-D Shapes.”)

Measuring squares

The letter s represents the length of a square's side. For example, if the side of a square is 3 inches, then you say s = 3 in. Finding the perimeter (P) of a square is simple: Just multiply the length of the side by 4. Here's the formula for the perimeter of a square:

For example, if the length of the side is 3 inches, substitute 3 inches for s in the formula:

Finding the area (A) of a square is also easy: Just multiply the length of the side by itself — that is, take the square of the side. Here are two ways of writing the formula for the area of a square (s² is pronounced “s squared”):

For example, if the length of the side is 3 inches, then you get the following:

Working with rectangles

The long side of a rectangle is called the length, or l for short. The short side is called the width, or w for short. For example, in a rectangle whose sides are 5 and 4 feet long, l = 5 ft. and w = 4 ft.

Because a rectangle has two lengths and two widths, you can use the following formula for the perimeter of a rectangle:

Calculate the perimeter of a rectangle whose length is 5 yards and whose width is 4 yards as follows:

The formula for the area of a rectangle is:

So here's how you calculate the area of the same rectangle:

Calculating with rhombuses

As with a square, use s to represent the length of a rhombus's side. But another key measurement for a rhombus is its height. The height of a rhombus (h for short) is the shortest distance from one side to the opposite side. In Figure 16-12, s = 4 cm and h = 2 cm.

Illustration by Wiley, Composition Services Graphics

Figure 16-12: Measuring a rhombus.

The formula for the perimeter of a rhombus is the same as for a square:

Here's how you figure out the perimeter of a rhombus whose side is 4 centimeters:

To measure the area of a rhombus, you need both the length of the side and the height. Here's the formula:

So here's how you determine the area of a rhombus with a side of 4 cm and a height of 2 cm:

You can read 8 cm² as “8 square centimeters” or, less commonly, as “8 centimeters squared.”

Measuring parallelograms

The top and bottom sides of a parallelogram are called its bases (b for short), and the remaining two sides are its sides (s). And as with rhombuses, another important measurement of a parallelogram is its height (h), the shortest distance between the bases. So the parallelogram in Figure 16-13 has these measurements: b = 6 in., s = 3 in., and h = 2 in.

Illustration by Wiley, Composition Services Graphics

Figure 16-13: Measuring a parallelogram.

Each parallelogram has two equal bases and two equal sides. Therefore, here's the formula for the perimeter of a parallelogram:

To figure out the perimeter of the parallelogram in this section, just substitute the measurements for the bases and sides:

And here's the formula for the area of a parallelogram:

Here's how you calculate the area of the same parallelogram:

Measuring trapezoids

The parallel sides of a trapezoid are called its bases. Because these bases are different lengths, you can call them b₁ and b₂. The height (h) of a trapezoid is the shortest distance between the bases. Thus, the trapezoid in Figure 16-14 has these measurements: b₁ = 2 in., b₂ = 3 in., and h = 2 in.

Illustration by Wiley, Composition Services Graphics

Figure 16-14: Measuring a trapezoid.

Because a trapezoid can have sides of four different lengths, you really don't have a special formula for finding the perimeter of a trapezoid. Just add up the lengths of its sides, and you get your answer.

Here's the formula for the area of a trapezoid:

So here's how to find the area of the pictured trapezoid:

Measuring triangles

In this section, I discuss how to measure the perimeter and area of all triangles. Then I show you a special feature of right triangles that allows you to measure them more easily.

Finding the perimeter and area of a triangle

Mathematicians have no special formula for finding the perimeter of a triangle — they just add up the lengths of the sides.

To find the area of a triangle, you need to know the length of one side — the base (b for short) — and the height (h). Note that the height forms a right angle with the base. Figure 16-15 shows a triangle with a base of 5 cm and a height of 2 cm:

Illustration by Wiley, Composition Services Graphics

Figure 16-15: The base and height of a triangle.

Here's the formula for the area of a triangle:

So here's how to figure out the area of a triangle with a base of 5 cm and a height of 2 cm:

Lessons from Pythagoras: Finding the third side of a right triangle

The long side of a right triangle (c) is called the hypotenuse, and the two short sides (a and b) are called the legs (see Figure 16-16). The most important right triangle formula is the Pythagorean theorem:

Illustration by Wiley, Composition Services Graphics

Figure 16-16: The hypotenuse and legs of a right triangle.

This formula allows you to find the hypotenuse of a triangle, given only the lengths of the legs. For example, suppose the legs of a triangle are 3 and 4 units. Here's how to use the Pythagorean theorem to find the length of the hypotenuse:

So when you multiply c by itself, the result is 25. Therefore,

· c = 5

The length of the hypotenuse is 5 units.

Going ’round in circles

The center of a circle is a point that's the same distance from any point on the circle itself. This distance is called the radius of the circle, or r for short. And any line segment from one point on the circle through the center to another point on the circle is called a diameter, or d for short. See Figure 16-17.

Illustration by Wiley, Composition Services Graphics

Figure 16-17: Deciphering the parts of a circle.

As you can see, the diameter of any circle is made up of one radius plus another radius — that is, two radii (pronounced ray-dee-eye). This concept gives you the following handy formula:

For example, given a circle with a radius of 5 millimeters, you can figure out the diameter as follows:

Because the circle is an extra-special shape, its perimeter (the length of its “sides”) has an extra-special name: the circumference (C for short). Early mathematicians went to a lot of trouble figuring out how to measure the circumference of a circle. Here's the formula they hit upon:

Note: Because 2 × r is the same as the diameter, you also can write the formula as C = π × d.

The symbol π is called pi (pronounced “pie”). It's just a number whose approximate value is as follows (the decimal part of pi goes on forever, so you can't get an exact value for pi):

So given a circle with a radius of 5 mm, you can figure out the approximate circumference:

The formula for the area (A) of a circle also uses π:

Here's how to use this formula to find the approximate area of a circle with a radius of 5 mm:

Spacing out: Measuring in three dimensions

In three dimensions, the concepts of area has to be tweaked a little. Recall that, in 2-D, the area of a shape is the measurement of what's inside the shape. In 3-D, what's inside a solid is called its volume.

The volume (V) of a solid is a measurement of the space it occupies, as measured in cubic units, such as cubic inches (in.³), cubic feet (ft.³), cubic meters (m³), and so forth. (For info on measurement, flip to Chapter 15.) Finding the volume of solids, however, is something mathematicians love for you to know. In the next sections, I give you the formulas for finding the volumes of a variety of solids.

Cubes

The main measurement of a cube is the length of its side (s). Using this measurement, you can find out the volume of a cube, using the following formula:

So if the side of a cube is 5 meters, here's how you figure out its volume:

You can read 125 m³ as “125 cubic meters” or, less commonly, as “125 meters cubed.”

Boxes (rectangular solids)

The three measurements of a box (or rectangular solid) are its length (l), width (w), and height (h). The box pictured in Figure 16-18 has the following measurements: l = 4 m, w = 3 m, and h = 2 m.

Illustration by Wiley, Composition Services Graphics

Figure 16-18: Measuring a box.

You can find the volume of a box, using the following formula:

So here's how to find the volume of the box pictured in this section:

Prisms

Finding the volume of a prism (see prisms in Figure 16-7) is easy if you have two measurements. One measurement is the height (h) of the prism. The second is the area of the base (A_b). The base is the polygon that extends vertically from the plane. (In “2-D: Measuring on the flat,” earlier, I show you how to find the area of a variety of shapes.)

Here's the formula for finding the volume of a prism:

For example, suppose a prism has a base with an area of 5 square centimeters and a height of 3 centimeters. Here's how you find its volume:

Notice that the units of measurements (cm² and cm) are also multiplied, giving you a result of cm³.

Cylinders

You find the volume of cylinders the same way you find the area of prisms — by multiplying the area of the base (A_b) by the cylinder's height (h):

Suppose you want to find the volume of a cylindrical can whose height is 4 inches and whose base is a circle with a radius of 2 inches. First, find the area of the base by using the formula for the area of a circle:

This area is approximate because I use 3.14 as an approximate value for π. (Note: In the preceding problem, I use equals signs when a value is equal to whatever comes right before it, and I use “approximately equal to” signs [≈] when I round.)

Now use this area to find the volume of the cylinder:

Notice how multiplying square inches (in.² ) by inches gives a result in cubic inches (in.³ ).