Geometry Articles

Cheat Sheet / Updated 03-09-2022

Geometry is full of formulas, properties, and theorems. You can become successful in geometry by remembering the most important ones and learning how to apply them. Use this reference sheet as you practice various geometry problems to grow your knowledge and skills.

Cheat Sheet / Updated 02-08-2022

Successfully understanding and studying geometry involves using strategies for your geometry proofs, knowing important equations, and being able to identify commonly used geometry symbols.

Article / Updated 12-21-2021

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. Once you get to know the types of angles and how to measure and create your own, you'll have picked up valuable geometry skills that will help you prove even the most complex geometric puzzles. To do both tasks, you use a protractor, a very useful tool to keep around (see Figure 1). When choosing a protractor, try to find one made of clear plastic. Figuring out the measure of an angle is easier because you can see the line for the angle through the protractor. The breeds of angles Several different angle breeds, or types, exist. You can figure out what breed of angle you have by its measure. The most common measure of an angle is in degrees. Here is a brief introduction to the four types of angles: Right angle. With this angle, you can never go wrong. The right angle is one of the most easily recognizable angles. It's in the form of the letter L, and it makes a square corner (see Figure 2). It has a measure of 90 degrees. Straight angle. You know what? It's actually a straight line. Most people don't even think of this type as an angle, but it is. A straight angle is made up of opposite rays or line segments that have a common endpoint (see Figure 3). This angle has a measure of 180 degrees. Right and straight angles are pretty easy to spot just by looking at them, but never jump to conclusions about the measure of an angle. Being cautious is best. If the info isn't written on the page, don't assume anything. Measure. Acute angle. It's the adorable angle. Actually, it's just a pinch. It's any angle that measures more than 0 degrees but less than 90 degrees. An acute angle falls somewhere between nonexistent and a right angle (see Figure 4). Obtuse angle. This type is just not as exciting as an acute angle. It's measure is somewhere between a right angle and a straight angle (see Figure 5). It is a hill you must climb, a mountain for you to summit. It has a measure of more than 90 degrees but less than 180 degrees. Measuring angles Angles are most commonly measured by degrees, but for those of you who are sticklers for accuracy, even smaller units of measure can be used: minutes and seconds. These kinds of minutes and seconds are like the ones on a clock — a minute is bigger than a second. So think of a degree like an hour, and you've got it down: One degree equals 60 minutes. One minute equals 60 seconds. Before measuring an angle, spec it out and estimate which type you think it is. Is it a right angle? A straight angle? Acute or obtuse? After you estimate it, then measure the angle. Follow these steps: Place the notch or center point of your protractor at the point where the sides of the angle meet (the vertex). Place the protractor so that one of the lines of the angle you want to measure reads zero (that's actually 0°). Using the zero line isn't necessary because you can measure an angle by getting the difference in the degree measures of one line to the other. It's easier, however, to measure the angle when one side of it is on the zero line. Having one line on the zero line allows you to read the measurement directly off the protractor without having to do more math. (But if you're up for the challenge, knock yourself out.) Read the number off the protractor where the second side of the angle meets the protractor. Some more advice: Make sure that your measure is close to your estimate. Doing so tells you whether you chose the proper scale. If you were expecting an acute angle measure but got a seriously obtuse measure, you need to rethink the scale you used. Try the other one. If the sides of your angle don't reach the scale of your protractor, extend them so that they do. Doing so increases the accuracy of your measure. Remember that the measure of an angle is always a positive number. So what do you do if your angle doesn't quite fit on the protractor's scale? Look at Figure 6 for an example. The angle in this figure has a measure of greater than 180°. Now what? Sorry, but in this case, you're going to have to expend a little extra energy. Yes, you have to do some math. These angles are known as reflex angles and they have a measure of greater than 180°. Draw a line so that you have a straight line (see the extended dots on Figure 6). The measure of this portion of the angle is 180° because it's a straight angle. Now measure the angle that is formed by the extension line you just made and the second side of the original angle you want to measure. (If you get confused, just look at Figure 6.) Once you have the measure of the second angle, add that number to 180. The result is the total number of degrees of the angle. In Figure 6, 180° + 45° = 225°.

Article / Updated 09-17-2021

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. 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. Note that to name a minor arc, you use its two endpoints; to name a major arc, you use its two endpoints plus any point along the arc. Central angle: A central angle is an angle whose vertex is at the center of a circle. The two sides of a central angle are radii that hit the circle at the opposite ends of an arc—or as mathematicians say, the angle intercepts the arc. The measure of an arc is the same as the degree measure of the central angle that intercepts it. Determining the length of an arc An arc’s length means the same commonsense thing length always means — you know, like the length of a piece of string (with an arc, of course, it’d be a curved piece of string). Make sure you don’t mix up arc length with the measure of an arc which is the degree size of its central angle. A circle is 360° all the way around; therefore, if you divide an arc’s degree measure by 360°, you find the fraction of the circle’s circumference that the arc makes up. Then, if you multiply the length all the way around the circle (the circle’s circumference) by that fraction, you get the length along the arc. So finally, here’s the formula you’ve been waiting for. Arc length: Its degree measure is 45° and the radius of the circle is 12, so here’s the math for its length: Pretty simple, eh?

Article / Updated 07-12-2021

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). If both pairs of opposite sides of a quadrilateral are congruent, then it’s a parallelogram (converse of a property). Tip: To get a feel for why this proof method works, take two toothpicks and two pens or pencils of the same length and put them all together tip-to-tip; create a closed figure, with the toothpicks opposite each other. The only shape you can make is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then it’s a parallelogram (converse of a property). If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram (converse of a property). Tip: Take, say, a pencil and a toothpick (or two pens or pencils of different lengths) and make them cross each other at their midpoints. No matter how you change the angle they make, their tips form a parallelogram. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it’s a parallelogram (neither the reverse of the definition nor the converse of a property). Tip: Take two pens or pencils of the same length, holding one in each hand. If you keep them parallel, no matter how you move them around, you can see that their four ends form a parallelogram. The preceding list contains the converses of four of the five parallelogram properties. If you’re wondering why the converse of the fifth property (consecutive angles are supplementary) isn’t on the list, you have a good mind for details. The explanation, essentially, is that the converse of this property, while true, is difficult to use, and you can always use one of the other methods instead.

Article / Updated 07-09-2021

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. Each lower base angle is supplementary to the upper base angle on the same side. The properties of the isosceles trapezoid are as follows: The properties of a trapezoid apply by definition (parallel bases). The legs are congruent by definition. The lower base angles are congruent. The upper base angles are congruent. Any lower base angle is supplementary to any upper base angle. The diagonals are congruent. The supplementary angles might be the hardest property to spot in the diagrams above. Because of the parallel sides, consecutive angles are same-side interior angles and are thus supplementary. (All the special quadrilaterals except the kite, by the way, contain consecutive supplementary angles.) Here’s an isosceles trapezoid proof for you: Statement 1: Reason for statement 1: Given. Statement 2: Reason for statement 2: The legs of an isosceles trapezoid are congruent. Statement 3: Reason for statement 3: The upper base angles of an isosceles trapezoid are congruent. Statement 4: Reason for statement 4: Reflexive Property. Statement 5: Reason for statement 5: Side-Angle-Side, or SAS (2, 3, 4) Statement 6: Reason for statement 6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Statement 7: Reason for statement 7: If angles are congruent, then so are sides.

Article / Updated 07-08-2021

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. Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side. Interior and exterior angle formulas: The sum of the measures of the interior angles of a polygon with n sides is (n – 2)180. The measure of each interior angle of an equiangular n-gon is If you count one exterior angle at each vertex, the sum of the measures of the exterior angles of a polygon is always 360°. Check here for more practice.

Article / Updated 07-08-2021

The three special parallelograms — rhombus, rectangle, and square — are so-called because they’re special cases of the parallelogram. (In addition, the square is a special case or type of both the rectangle and the rhombus.) The three-level hierarchy you see with in the above quadrilateral family tree works just like A dog is a special type of a mammal, and a Dalmatian is a special type of a dog. Here are the properties of the rhombus, rectangle, and square. Note that because these three quadrilaterals are all parallelograms, their properties include the parallelogram properties. The rhombus has the following properties: All of the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary). All sides are congruent by definition. The diagonals bisect the angles. The diagonals are perpendicular bisectors of each other. The rectangle has the following properties: All of the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals bisect each other). All angles are right angles by definition. The diagonals are congruent. The square has the following properties: All of the properties of a rhombus apply (the ones that matter here are parallel sides, diagonals are perpendicular bisectors of each other, and diagonals bisect the angles). All of the properties of a rectangle apply (the only one that matters here is diagonals are congruent). All sides are congruent by definition. All angles are right angles by definition. Now try working through a problem. Given the rectangle as shown, find the measures of angle 1 and angle 2: Here’s the solution: MNPQ is a rectangle, so angle Q = 90°. Thus, because there are 180° in a triangle, you can say Now plug in 14 for all the x’s. Now find the perimeter of rhombus RHOM. Here’s the solution: All the sides of a rhombus are congruent, so HO equals x + 2. And because the diagonals of a rhombus are perpendicular, triangle HBO is a right triangle. You finish with the Pythagorean Theorem: Combine like terms and set equal to zero: Factor: (x – 3)(x + 1) = 0 Use Zero Product Property: x – 3 = 0 or x + 1 = 0 x = 3 or x = –1 You can reject x = –1 because that would result in triangle HBO having legs with lengths of –1 and 0.

Article / Updated 07-07-2021

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 four corners (like triangle SUE in the figure) that you cut off the square to turn it into an octagon are 45°- 45°- 90° triangles (you can prove that to yourself if you feel like it). So all you have to do to get the area of the octagon is to calculate the area of the square and then subtract the four corner triangles. Piece o’ cake. But first, here are two great tips for this and other problems. For problems involving regular octagons, 45°- 45°- 90° triangles can come in handy. Add segments to the diagram to get one or more 45°- 45°- 90° triangles and some squares and rectangles to help you solve the problem. Think outside the box. It’s easy to get into the habit of looking only inside a figure because that suffices for the vast majority of problems. But occasionally (like in this problem), you need to break out of that rut and look outside the perimeter of the figure. Okay, so here’s what you do. You’re given that the octagon’s sides have a length of 6. Consider side EU. Not just calculate the area of the square and the area of a single corner triangle: To finish, subtract the total area of the four corner triangles from the area of the square:

Article / Updated 07-07-2021

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. Chord: A segment that connects two points on a circle is called a chord. Diameter: A chord that passes through a circle’s center is a diameter of the circle. A circle’s diameter is twice as long as its radius. The above figure shows circle O.