Geometry For Dummies
Overview
Hit the geometry wall? Get up and running with this no-nonsense guide!
Does the thought of geometry make you jittery? You're not alone. Fortunately, this down-to-earth guide helps you approach it from a new angle, making it easier than ever to conquer your fears and score your highest in geometry. From getting started with geometry basics to making friends with lines and angles, you'll be proving triangles congruent, calculating circumference, using formulas, and serving up pi in no time.
Geometry is a subject full of mathematical richness and beauty. But it's a subject that bewilders many students because it's so unlike the math they've done before—it requires the use of deductive logic in formal proofs. If you're having a hard time wrapping your mind around what that even means, you've come to the right place! Inside, you'll find out how a proof's chain of logic works and even discover some secrets for getting past rough spots along the way. You don't have to be a math genius to grasp geometry, and this book helps you get un-stumped in a hurry!
- Find out how to decode complex geometry proofs
- Learn to reason deductively and inductively
- Make sense of angles, arcs, area, and more
- Improve your chances of scoring higher in your geometry class
There's no reason to let your nerves get jangled over geometry—your understanding will take new shape with the help of Geometry For Dummies.
Read More
About The Author
Mark Ryan is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation. Mark is the author of Calculus For Dummies, Calculus Workbook For Dummies, and Geometry Workbook For Dummies.
Sample Chapters
geometry for dummies
CHEAT SHEET
Successfully understanding and studying geometry involves using strategies for your geometry proofs, knowing important equations, and being able to identify commonly used geometry symbols.Geometry formulas, theorems, properties, and moreWhat follows are over three dozen of the most important geometry formulas, theorems, properties, and so on that you use for calculations.
View Cheat Sheet
View Cheat Sheet
HAVE THIS BOOK?
View Book Companion Site
Access practice tests
Additional Book Resources
Articles from
the book
You can add bubbles and arrows to a proof solution to show the connections between the statements and the reasons. Although you won't be asked to do this when you solve a proof, it's a great way to help you understand how proofs work.Here's an example of a bubble proof. Follow the arrows from bubble to bubble.
To calculate the surface area of a cone, you need to add the area of the cone’s base to its lateral area. (The lateral area is a fancy name for the area of the surface that connects the base to the peak; in other words, everything but the base.) A cone with its height and slant height.
The following formula gives you the surface area of a cone.
A cylinder has two congruent bases, which makes it easy to calculate its surface area: you simply find the area of one base and double that value; then you add the cylinder's lateral area (or lateral "rectangle"). A cylinder with its bases and lateral "rectangle."
As shown here, the lateral area of a cylinder is the area of the sides of the cylinder—namely, the area of everything but the cylinder's bases.
Because prisms have two congruent bases, it's easy to calculate their surface area: first, you find the area of one base and double that value; then, you add the prism's lateral area.
The lateral area of a prism is the area of its sides—namely, the area of everything but the prism's bases. Although the bases of a prism can be any shape, the lateral area is always made up of rectangles.
You can find the surface area of a pyramid by adding the lateral area of the pyramid (basically, four triangles) to the area of the pyramid's base (a square). A pyramid with its height and slant heights.The following formula gives you the surface area of a pyramid.The lateral area of a pyramid is the area of the surface that connects the base to the peak (it's the area of everything but the base).
A sphere doesn't have any flat sides—in fact, its entire surface is one big curve. So finding its surface area must be impossible, right? Actually, it's surprisingly easy. You only need one measurement: its radius.
The radius of a sphere goes from its center to its surface.
Use the following formula for the surface area of a sphere.
When you want to compute the volume of a cone, you need only two things: its height and the radius of its base. Even if you are given its slant height instead of its vertical height, you can still find the volume; you just need to include an extra step. A cone with its height and slant height.
A cone is a solid figure with a rounded base and a rounded lateral surface that connects the base to a single point.
The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow.To calculate the volume of a cylinder, you need to know its height and the area of its base. Because a cylinder is a flat-top figure (a solid with two congruent, parallel bases), the base can be either the top or bottom.
To calculate the volume of a prism, you first need to know its height and the area of its base (top or bottom, it doesn't matter—they're parallel and congruent). A prism with its bases and lateral rectangles.
The volume of a prism is given by the following formula:
An ordinary box is a special case of a prism, so you can use this volume formula for a box, but you probably already know the easier way to compute a box's volume:(Because the length times the width gives you the area of the base, these two methods really amount to the same thing.
Pyramids are full of congruent right triangles. So, to find the volume of a pyramid, it's just a matter of using the Pythagorean Theorem or your knowledge of special right triangles.
A pyramid is a solid figure with a polygonal base and edges that extend up from the base to meet at a single point. The corners of a pyramid are called vertices, the segments that connect the vertices are called edges, and the flat sides are called faces.
To calculate the volume of a sphere, you need to know its radius—the distance from its center to its surface. Then, you simply plug that number into the following formula:Let's try a problem. Because a simple problem where you're given a sphere's radius is too easy, the following sphere problem involves a cube (just to make things interesting!
Geometry
Say you're taking a geometry test, and one question gives you a line along with a point that is not on that line. You're told to construct a line perpendicular to the given line that also passes through the point.Can it be done? Well, yes—but you'll need a compass to make it happen.The following figure will help guide you through a construction.
Geometry
Say you're given a line with a point somewhere along it, and you have to construct a second line perpendicular to the first one through that point. Can it be done? Absolutely! The following example will show you how.As you work through this construction, you can refer back to this figure: Constructing a perpendicular line through a point on a line.
Geometry
To construct a perpendicular bisector of a segment, you use your compass to locate two points that are each equidistant from the segment's endpoints and then finish with your straightedge. The following example shows you how it's done.This figure should help as you work through the construction process: Constructing a perpendicular bisector.
A geometric reflection works like a reflection in a mirror. It's also the basic building block for other types of isometric transformations. Whether you're creating a translation, rotation, or glide reflection, you always start with a reflection. A triangle's reflection in a mirror.The figure shows someone in front of a mirror looking at the reflection of a triangle that's on the floor in front of the mirror.
A good way to begin any geometric proof is to make a game plan, or rough outline, of how you'd do the proof. The formal way of writing out a two-column proof can be difficult, especially at first—almost like learning a foreign language.Writing a proof is easier if you break it into two shorter, more-manageable pieces.
Say you're given a line segment and you have to find the coordinates of its midpoint. What's the best way to do this? The midpoint formula!The way the midpoint formula works is very simple: It takes the average of the x-coordinates of the segment's endpoints and the average of the y-coordinates of the endpoints.
Geometry
If you can't immediately see which Pythagorean triple family a triangle belongs to, don't worry: you can always use the following step-by-step method to pick the family and find the missing side.For example, say you're given the following tricky triangle: You can use a ratio to figure out what family this triangle belongs to.
A rotation is a transformation in which the pre-image figure rotates or spins to the location of the image figure. With all rotations, there's a single fixed point—called the center of rotation—around which everything else rotates.This point can be inside the figure, in which case the figure stays where it is and just spins.
Geometry
If you're given two points, and you're asked to find the locus of points equidistant from these two points, you'll always find the same thing: that the locus of points is actually the perpendicular bisector of the segment that joins the two points.If that sounds a little technical, don't worry—the following example will make everything clear!
Geometry
A glide reflection involves three reflections, and so it can be challenging to find the location of its main reflecting line. Fortunately, there's a handy theorem that you can use for just that purpose.
The main reflecting line of a glide reflection: In a glide reflection, the midpoints of all segments that connect pre-image points with their image points lie on the main reflecting line.
If you want to work with multiple-plane proofs, you first have to know how to determine a plane. Determining a plane is the fancy, mathematical way of saying "showing you where a plane is."There are four ways to determine a plane:
Three non-collinear points determine a plane. This statement means that if you have three points not on one line, then only one specific plane can go through those points.
Successfully understanding and studying geometry involves using strategies for your geometry proofs, knowing important equations, and being able to identify commonly used geometry symbols.Geometry formulas, theorems, properties, and moreWhat follows are over three dozen of the most important geometry formulas, theorems, properties, and so on that you use for calculations.
Geometry
What follows are over three dozen of the most important geometry formulas, theorems, properties, and so on that you use for calculations. If you get stumped while working on a geometry problem and can’t come up with a formula, this is the place to look.
Triangle stuff
Sum of the interior angles of a triangle: 180º
Area:
Hero’s area formula: , where a, b, and c are the lengths of the triangle’s sides and (S is the semiperimeter, half the perimeter)
Area of an equilateral triangle: , where s is a side of the triangle
The Pythagorean Theorem: a2 + b2 + c2, where a and b are the legs of a right triangle and c is the hypotenuse
Common Pythagorean triples (side lengths in right triangles):
3–4–5
5–12–13
7–24–25
8–15–17
Ratios of the sides in special right triangles:
The sides opposite the angles in a 45º–45º–90º triangle are in the ratio of .
Geometry
A translation—probably the simplest type of figure transformation in coordinate geometry—is one in which a figure just slides straight to a new location without any tilting or turning. As you can see, a translation doesn't change a figure's orientation. A trapezoid before and after a translation.It may seem a bit surprising, but instead of sliding a figure to a new location, you can achieve the same end result by reflecting the figure over one line and then over a second line.
A glide reflection is just what it sounds like: You glide a figure (that's just another way of saying slide or translate) and then reflect it over a reflecting line. Or you can reflect the figure first and then slide it; the result is the same either way. The footprints are glide reflections of each other.A glide reflection is also called a walk because it looks like the motion of two feet, as shown here.
Geometry
When you reflect a shape in coordinate geometry, the reflected shape remains congruent to the original, but something changes. That something is the new shape's orientation.For example, as you can see in the image, the triangle in the mirror is flipped over compared with the real triangle. A triangle's reflection in a mirror.
Geometry
There are three power theorems you can use to solve all sorts of geometry problems involving circles: the chord-chord power theorem, the tangent-secant power theorem, and the secant-secant power theorem.All three power theorems involve an equation with a product of two lengths (or one length squared) that equals another product of lengths.
To bisect an angle, you use your compass to locate a point that lies on the angle bisector; then you just use your straightedge to connect that point to the angle's vertex.Try an example.Refer to the figure as you work through this construction:
Open your compass to any radius r, and construct arc (K, r) intersecting the two sides of angle K at A and B.
Geometry
You can calculate the area of a regular octagon with the standard regular polygon method, but there’s a nifty alternative method based on the fact that a regular octagon is a square with its four corners cut off.For example, here’s how you’d find the area of EIGHTPLU in the figure below, given that it’s a regular octagon with sides of length 6.
The key to copying a given line segment is to open your compass to the length of the segment; then, using that amount of opening, you can mark off another segment of the same length.Here's the proof diagram. Copying a segment.This figure shows the solution; refer to it as you work through the following steps.
Using your straightedge, draw a working line, l, with a point P anywhere on it.
When you copy a triangle, the idea is to use your compass to “measure” the lengths of the three sides of the given triangle and then make another triangle with sides congruent to the sides of the original triangle.The fact that this method works is related to the SSS method of proving triangles congruent.
Here is the proof diagram.
The basic idea behind copying a given angle is to use your compass to sort of measure how wide the angle is open; then you create another angle with the same amount of opening.Here's the proof diagram. Copying an angle.Refer to the figure as you work through these steps:
Draw a working line, l, with point B on it.
When you do an analytic proof, your first step is to draw a figure in the coordinate system and label its vertices. This figure will make the algebra part easier, when you have to prove something about the figure.Here's an example. Say you're given the following proof: First, prove analytically that the midpoint of the hypotenuse of a right triangle is equidistant from the triangle's three vertices, and then show analytically that the median to this midpoint divides the triangle into two triangles of equal area.
When you create a reflection of a figure, you use a special line, called (appropriately enough) a reflecting line, to make the transformation. In coordinate geometry, the reflecting line is indicated by a lowercase l. Reflecting triangle PQR over line l switches the figure's orientation.This figure illustrates an important property of reflecting lines: If you form segment RR' by connecting pre-image point R with its image point R' (or P with P' or Q with Q'), the reflecting line, l, is the perpendicular bisector of segment RR'.
If you're given two points on the coordinate plane, you can find the locus of points that creates a right angle to those two points. By using the following steps, you'll soon see an interesting pattern that may, in fact, surprise you.Here's a problem to solve: Given points P and R, what's the locus of points Q such that angle PQR is a right angle?
Geometry
A circle's central angles and the arcs that they cut out are part of many circle proofs. They also come up in many area problems. The following figure shows how an angle and an arc are interrelated. A 60-degree central angle cuts out a 60-degree arc.
Arc: An arc is simply a curved piece of a circle. Any two points on a circle divide the circle into two arcs: a minor arc (the smaller piece) and a major arc (the larger)—unless the points are the endpoints of a diameter, in which case both arcs are semicircles.
When you work with circles, there are three straight-line components that you need to be able to identify: radii, chords, and diameters.
Radius: A circle’s radius — the distance from its center to a point on the circle — tells you the circle’s size. In addition to being a measure of distance, a radius is also a segment that goes from a circle’s center to a point on the circle.
Geometry
There are five ways in which you can prove that a quadrilateral is a parallelogram. The first four are the converses of parallelogram properties (including the definition of a parallelogram). Make sure you remember the oddball fifth one — which isn’t the converse of a property — because it often comes in handy:
If both pairs of opposite sides of a quadrilateral are parallel, then it’s a parallelogram (reverse of the definition).
Although it sounds like a top-secret government department, CPCTC is actually just an acronym for a statement about congruent triangles: Corresponding Parts of Congruent Triangles are Congruent.As you'll see in the following example, CPCTC is very useful when working with proofs. But first, you'll need the following property to do the problem.
CSSTP is an acronym that represents a simple but useful truth: In similar triangles, the Corresponding Sides of Similar Triangles are Proportional. CSSTP proofs often involve an odd step at the end where you have to prove that one product of sides equals another product of sides.You'll see what this means in the following problem:Here's the proof diagram.
In two-dimensional locus problems, all the points in the locus solution lie in a plane. This is usually, but not always, the same plane as the given geometric object.You can use the following four-step solution method to solve a 2-D problem.Start with a problem: What's the locus of all points 3 units from a given circle whose radius is 10 units?
When you solve an analytic proof, this involves using algebra. You can use analytic proofs to prove different properties; for example, you can prove the property that the diagonals of a parallelogram bisect each other, or that the diagonals of an isosceles trapezoid are congruent.
Before solving a proof, it's useful to draw your figure in the coordinate system and label its vertices.
Geometry
You can apply equations and algebra (that is, use analytic methods) to circles that are positioned in the x-y coordinate system. For example, there's a nice analytic connection between the circle equation and the distance formula because every point on a circle is the same distance from its center.
Here are the circle equations:
Circle centered at the origin, (0, 0),
x2 + y2 = r2
where r is the circle's radius.
Geometry
When a line is perpendicular to a plane, you can use this perpendicularity in two-column proofs. You simply apply the following definition and theorem of line-plane perpendicularity.
Line-Plane perpendicularity definition: Saying that a line is perpendicular to a plane means that the line is perpendicular to every line in the plane that passes through its foot.
With the side-splitter theorem, you draw one parallel line that divides a triangle's sides proportionally. With the extension of this theorem, you can draw any number of parallel lines that cut any lines (not just a triangle's sides) proportionally.
Extension of the Side-Splitter Theorem: If three or more parallel lines are intersected by two or more transversals, the parallel lines divide the transversals proportionally.
Everything you need to know about a polygon doesn’t necessarily fall within its sides. You may need to find exterior angles as well as interior angles when working with polygons:
Interior angle: An interior angle of a polygon is an angle inside the polygon at one of its vertices. Angle Q is an interior angle of quadrilateral QUAD.
On a map, you trace your route and come to a fork in the road. Two diverging roads split from a common point and form an angle. The point at which the roads diverge is the vertex. An angle separates the area around it, known in geometry as a plane, into two regions. The points inside the angle lie in the interior region of the angle, and the points outside the angle lie in the exterior region of the angle.
Knowing how to write two-column geometry proofs provides a solid basis for working with theorems. Practicing these strategies will help you write geometry proofs easily in no time:
Make a game plan. Try to figure out how to get from the givens to the prove conclusion with a plain English, commonsense argument before you worry about how to write the formal, two-column proof.
A rotation is what you'd expect—it's a geometric transformation in which the pre-image figure rotates or spins to the location of the image figure.With all rotations, there's a single fixed point—called the center of rotation—around which everything else rotates. This point can be inside the figure, in which case the figure stays where it is and just spins.
Geometry
When dealing with geometry problems where lines are tangent to circles, you can use a walk-around approach to solve them. First, though, you need to be familiar with the following theorem.
Dunce Cap Theorem: If two tangent segments are drawn to a circle from the same external point, then they're congruent. You can think of this as the dunce cap theorem because that's what the diagram for it looks like (although don't go searching for that name in a geometry book—you won't have much luck!
Geometry
A trapezoid is a quadrilateral (a shape with four sides) with exactly one pair of parallel sides (the parallel sides are called bases). The following figure shows a trapezoid to the left, and an isosceles trapezoid on the right.
The properties of the trapezoid are as follows:
The bases are parallel by definition.
In coordinate geometry, there are different line equations you can use, depending on whether a line is horizontal, vertical, or at an angle, and whether you know the line's y-intercept.
Here are the basic forms for equations of lines:
Slope-intercept form. Use this form when you know (or can easily find) a line's slope and its y-intercept (the point where the line crosses the y-axis).
Geometry
The slope of a line on the coordinate plane basically tells you how steep the line is. If you know the rise and run of a line, you can calculate its slope using the slope formula.
Slope formula: The slope of a line containing two points, (x1, y1) and (x2, y2), is given by the following formula (a line's slope is often represented by the letter m):
Note: It doesn't matter which points you designate as (x1, y1) and (x2, y2); the math works out the same either way.
If you need a quick refresher about how the x-y coordinate system works, you've come to the right place. Let's start with the following figure, which shows you the lay of the land of the coordinate plane. The x-y coordinate system.Here's the lowdown on the coordinate plane you see in the figure:
The horizontal axis, or x-axis, goes from left to right and works exactly like a regular number line.
If two points in the x-y coordinate system are located diagonally from each other, you can use the distance formula to find the distance between them. As you will see, this distance is also the length of a hypotenuse.
Distance formula: To calculate diagonal distances, mathematicians whipped up the distance formula, which gives the distance between two points (x1, y1) and (x2, y2):
Note: Like with the slope formula, it doesn't matter which point you call (x1, y1) and which you call (x2, y2).
Geometry
When writing a two-column proof, you need to spell out every little step as if you had to make the logic clear to a computer. To do this, you can use if-then logic to move from the givens to the final conclusion.The following example starts with 'given' information, and shows the first few lines of the proof.And here's the proof diagram.
Geometry
When lines and planes are perpendicular and parallel, they have some interesting properties. You can use some of these properties in 3-D proofs that involve 2-D concepts, such as proving that you have a particular quadrilateral or proving that two triangles are similar.
Three parallel planes: If two planes are parallel to the same plane, then they're parallel to each other.
Load More Articles
Related Books
0:00
0:00
https://cdn.prod.website-files.com/6630d85d73068bc09c7c436c/69195ee32d5c606051d9f433_4.%20All%20For%20You.mp3
Frequently Asked Questions
No items found.
