Carla Kirkland

Cheat Sheet / Updated 09-12-2022

Before you get too excited, understand that the following information isn’t actually about how to cheat on the Praxis. It’s really about the most efficient ways to prepare for the exam. But “preparation sheet” doesn’t quite have the same ring to it. Besides, cheating is unnecessary if you know what you’re doing, and sometimes figuring out what to do is actually easier. As Bart Simpson once said after accidentally studying for a test, “It was like a whole new way to cheat!”

Article / Updated 05-01-2017

The ancient pyramids have mystified people for thousands of years, just as pyramid questions on the Praxis Core exam have mystified ill-prepared test-takers. You can avoid this dreaded curse by remembering two simple formulas for the surface area and volume of a pyramid. The first practice question asks you to find a pyramid's surface area, while the second question drops a pyramid on top of a cube, and asks for their composite (combined) volume. Practice questions The following pyramid has a rectangular base. What is the surface area of the pyramid? A. 1,215 ft.2 B. 1,980 ft.2 C. 1,125 ft.2 D. 450 ft.2 E. 900 ft.2 Refer to the following figure for the next question. The composite figure is formed by a square pyramid on top of a cube. The pyramid and cube share a base. The height of the pyramid is 9 miles, and the height of the cube is 9 miles. What is the volume of the composite figure? A. 729 cubic miles B. 972 cubic miles C. 81 cubic miles D. 648 cubic miles E. 1,458 cubic miles Answers and explanations The correct answer is Choice (A). The surface area of a pyramid is its lateral area plus its base area, or The base perimeter of the pyramid here is the sum of its side measures, so it's 90 ft. The slant height is given as 17 ft. The base area is the product of the length and width of the base, so it's 450 ft.2. With those measures, you can determine the surface area of the pyramid." The surface area of the pyramid is 1,215 ft.2. The correct answer is Choice (B). The volume of the composite figure is the sum of the volume of the cube and the volume of the pyramid. Find each separately and then add the volumes. The volume of the cube is the cube of its side measure: The volume of the cube is 729 cubic miles. The volume of the pyramid is a third of its base area times its height. Its base is a square, so you can square its side measure to find the base area: 92 = 81 The base area of the pyramid is 81 square miles. With that and the height of the pyramid, you can find the pyramid's volume: Notice that the volume of the pyramid is 1/3 times the volume of the cube. That's because they have the same base area and height. The sum of the volume of the cube and the volume of the pyramid is the sum of 729 cubic miles and 243 cubic miles: 729 + 243 = 972 The volume of the composite figure is 972 cubic miles.

Article / Updated 05-01-2017

The best cones are those filled with chocolate ice cream. The second-best—well, a distant second—are the ones you'll find on the Praxis Core exam. As you'll see in the following practice questions, you may be asked to calculate a cone's surface area (in this case, based on its lateral area and base area) or its volume (in this case, given its radius and slant height). Practice questions A cone has a lateral area of and a base area of . How many square centimeters is the surface area of the cone? Refer to the following figure for the next question. What is the volume of the cone? Answers and explanations The correct answer is Choice (E). The surface area of a cone is the sum of its lateral area (L) and base area. A cone has only one base, so you add B to the lateral area instead of 2B. The surface area of the cone is The correct answer is Choice (B). The volume of a cone is a third of the product of its base and its height. The height of this cone isn't given, but you can use the Pythagorean theorem to find it. The height, a radius, and the slant height form a right triangle in which the height and the radius are perpendicular and the slant height is the hypotenuse. The height of the cone is 24 m. That times a third of the base area is the volume of the cone: The volume of the cone is

Article / Updated 05-01-2017

You can think of a cylinder as a circle with attitude. If you encounter a cylinder problem on the Praxis Core exam, you can knock it down to size if you remember a few simple formulas. In the first practice question, you apply the surface area formula to a right cylinder. In the second question, you need to use the volume formula—and a little subtraction—to get the right answer. Practice questions Refer to the following figure for the first question. What is the surface area of the right cylinder? Refer to the following figure for the next question. In the diagram, a cylinder is on the interior of another cylinder. The bases of the interior cylinder are parts of the bases of the larger cylinder. The centers of the bases are shared by the cylinders. How much volume of the larger cylinder is outside of the smaller cylinder? Answers and explanations The correct answer is Choice (A). The radius of a cylinder, as with just a circle, is half its diameter. The radius for the cylinder here is half of 8 km, so it's 4 km. Use the surface area formula for a right prism, where SA is the surface area, P is the perimeter of the base (the circumference in this case), h is the height, and B is the base area: The surface area of the cylinder is 152 π km2. The correct answer is Choice (C). The amount of volume of the larger cylinder that is outside of the smaller cylinder is the larger cylinder's volume minus the smaller cylinder's volume. To find that difference, first find the larger cylinder's volume: The volume of the larger cylinder is Next, find the volume of the smaller cylinder: The volume of the larger cylinder is The difference between the larger cylinder's volume and the smaller cylinder's volume is

Article / Updated 05-01-2017

Prisms weren't invented just to divide light into beautiful rainbows; they're also useful for inducing painful headaches when you're asked to find their surface area or volume on the Praxis Core exam. Swallow a headache tablet and try the following practice questions. In the first one, you have to find the surface area of a right rectangular prism. In the second one, you're asked to find the volume of a triangular prism. Practice questions The following figure is a right rectangular prism. What is its surface area? A. 266 m2 B. 144 m2 C. 246 m2 D. 432 m2 E. 348 m2 A triangular prism has two triangular bases that both have an area of 19 square units. The prism has a height of 8 units. What is the volume of the prism? A. 604 cubic units B. 76 cubic units C. 27 cubic units D. 152 cubic units E. 228 cubic units Answers and explanations The correct answer is Choice (E). The surface of any right prism is Ph + 2B, which is the product of the perimeter of a base and its corresponding height, plus twice the area of the base. Any face of a right rectangular prism can be considered a base, and what you use for height must be the measure of a segment perpendicular to the face you decided to consider a base. For example, if you consider an 8 m-by-9 m face a base for the right rectangular prism in this problem, you must consider 6 m to be the height. In that case, the other 8 m-by-9 m face is also a base. The surface area of the right rectangular prism is 348 m2. The correct answer is Choice (D). The volume of any prism is Bh, or base area times the height: The volume of the prism is 152 cubic units.

Article / Updated 05-01-2017

Even when two shapes look identical, they may not be. For example, on the Praxis Core exam, you may encounter a question where two shapes look the same, but you're told they are "similar." What does this mean? What if they're labeled "congruent"? If you're not sure, the following practice questions (and their explanations) should refresh your memory. In the first question, you're shown two parallelograms that are similar, and you have to find their scale factor. The second question shows you two congruent triangles, and asks you to select all of the true statements about them. Practice questions Refer to the following figure for the first question. The two parallelograms in the following diagram are similar. Line BD corresponds to line FH, and line AB corresponds to EF. What is the measure of line EF? The two triangles in the following diagram are congruent. Which of these statements is true? Select all that apply. A. All pairs of corresponding angles are congruent. B. All pairs of corresponding sides are congruent. C. All corresponding parts are congruent. D. The two triangles' perimeters are equal. E. All pairs of corresponding angles are congruent, but none of the pairs of corresponding sides are necessarily congruent. Answers and explanations The correct answer is Choice (E). You can set up a proportion with ratios of corresponding side measures. Use a variable to represent the unknown measure: The side measure in question is 6 m. The correct answers are Choices (A, B, C, and D). If two triangles are congruent, they're identical; they're essentially the same triangle in two different places. By definition, all pairs of corresponding parts between two congruent triangles are congruent. That means their corresponding sides are congruent in every case, and their corresponding angles are congruent in every case. Because their corresponding sides are congruent, the sums of their sides are the same, so their perimeters are equal. Choice (E) isn't true, because all pairs of corresponding sides are congruent in all cases of congruent triangles. Choice (E) is true of similar triangles but not congruent triangles.

Article / Updated 05-01-2017

If you encounter a right triangle question on the Praxis Core exam, you can use good old Pythagoras' theorem to work out the answer. In some cases, you'll be able to skip the calculations and solve the problem using the properties of common right triangles! Both of the following practice questions can be worked out using the Pythagorean theorem—but if you know your common right triangles, you'll be done before you can say "Eureka!" Practice questions For right triangle ABC, what is the measure of side AB? A. 64 m B. 36 m C. 18 m D. 4 m E. 8 m A triangle has sides of 24 dm, 10 dm, and 26 dm. Is the triangle a right triangle? A. Yes B. No Answers and explanations The correct answer is Choice (E). For every right triangle, the square of the hypotenuse is equal to the sum of the squares of its legs. The legs are the sides that form the right angle, and the hypotenuse is the side that's across from the right angle. The formula is commonly represented as a2 + b2 = c2, where a and b are the legs and c is the hypotenuse. Fill in what is known and solve for what is unknown: AB is 8 m. Knowing the side lengths of common right triangles can shave off much-needed time. For example, well-known right triangles include the 3-4-5 triangle (and its multiples, such as the 6-8-10 and the 9-12-15), the 5-12-13 triangle, and the 8-15-17 triangle. Based on this information, you could have known the other side length would be 8 without using the Pythagorean theorem. The correct answer is Choice (A). If the triangle is a right triangle, the Pythagorean theorem applies to it — that is, the sum of the squares of the two leg measures equals the square of the hypotenuse. You can substitute the triangle's measures into a2 + b2 = c2 and see whether the equation works: The equation is true, so the triangle is a right triangle.

Article / Updated 05-01-2017

When you take the Praxis Core exam, it pays to have a well-rounded knowledge of circles—especially their area and circumference. In the following practice questions, you work both backwards (finding a circle's radius given its circumference) and forward (finding a circle's area given its radius). Practice questions A circle has a circumference of 20π in. What is the radius of the circle? A. 4.5 in. B. 15 in. C. 10 in. D. 20 in. E. 17.5 in. The two circles have congruent radii. If the radius of one circle is 3 m, what is the area of the other circle, rounded to the nearest hundredth? A. 6π m2 B. 18 π m2 C. 14.31 m2 D. 28.26 m2 E. 18.35 m2 Answers and explanations The correct answer is Choice (C). The circumference of a circle is 2 times pi times the radius. You can use the formula for circumference, fill in what you know, and solve for r, the radius of the circle: The radius of the circle is 10 in. The correct answer is Choice (D). The circles' radii are congruent, which means they have the same measure. Because one circle's radius is 3 m, the circle in question has a radius of 3 m. You can use the formula for the area of a circle: Because pi rounded to the nearest hundredth is 3.14, you can multiply 9 by 3.14: 9 × 3.14 = 28.26 The area of the circle, rounded to the nearest hundredth, is 28.26 m2.

Article / Updated 05-01-2017

Interior angle problems on the Praxis Core exam are pretty straightforward—well, most of the time. As you’ll see in the following practice questions, it’s also useful to have a good overall knowledge of angles for the exam. The formula for the sum of interior angles for a polygon is where s is the number of sides of the polygon. In the first practice question, you’re asked to find the missing interior angle in a quadrilateral. The second question gets a little trickier, because it also tests your knowledge of isosceles triangles as well as vertical angles. Practice questions Refer to this diagram to answer the following question. What is the value of k in the diagram? A. 15 B. 105 C. 285 D. 35 E. 75 What is the value of n in the following diagram? A. 80 B. 140 C. 120 D. 65 E. 40 Answers and explanations The correct answer is Choice (E). The sum of the interior angle measures of a quadrilateral (four-sided polygon) is 360 degrees. Therefore, the sum of 108, 101, 76, and k is 360. You can set up an equation and solve for k. The correct answer is Choice (E). If two sides of a triangle are congruent, the angles opposite those sides are congruent. Therefore, the two triangle angles that aren’t labeled with measures have the same measure. Their sum must be 80 degrees, because the sum of the interior angles of the triangle is 180 degrees and the labeled angle is 100 degrees: 180 – 100 = 80 Because the sum of the two angles is 80 degrees and the two angles have the same measure, each one has to be 40 degrees. One of them is a vertical angle to the n-degree angle, and vertical angles are congruent, so the n-degree angle is also 40 degrees. Therefore, the value of n is 40.

Article / Updated 05-01-2017

If you remember the properties of complementary and supplementary angles, this will definitely come in handy for the Praxis Core exam. If you don’t, check out the following tip and practice questions to refresh your memory. When two adjacent angles add up to 90 degrees (forming a right angle), they are complementary. When two adjacent angles add up to 180 degrees (forming a straight angle, or linear pair), they are supplementary. Both of the following practice questions contain diagrams. In the first one, you’re shown one angle in a linear pair, and you have to calculate the missing supplementary angle. In the second problem, you’re shown one angle in a right angle, and you have to find its missing complementary angle. Practice questions Use the following diagram to answer the first question. What is the measure of angle ABD? A. 121 degrees B. 31 degrees C. 59 degrees D. 61 degrees E. 149 degrees What is the value of x in the following diagram? A. 143 B. 53 C. 233 D. 37 E. 63 Answers and explanations The correct answer is Choice (C). Angle ABD and angle DBC form a linear pair, so they're supplementary. Therefore, the sum of their measures is 180 degrees. The measure that must be added to 121 degrees is 59 degrees because 180 – 121 = 59. Therefore, the measure of angle ABD is 59 degrees. The correct answer is Choice (B). You can work a problem like this by setting up an algebraic equation. Here, the sum of x and 37 must be 90 because the two angles are complementary. They form a right angle together, so the sum of their measures is 90 degrees.