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Lines and Angles

Lines and angles are the foundation of SAT geometry. Therefore, mastering their basic rules will make solving these questions, as well as related geometry questions, easier. With the knowledge you’ll gain from this chapter, you can quickly identify geometric relationships, build upon the information given in the question, and often bypass complex algebra.

Familiarity with angle types will often unlock information that is not explicitly given in a question. This makes getting to the answer much easier for even the toughest geometry questions. First, let’s take a look at the types of angles you should be able to recognize.

Angle Type	Angle Measurement	Example
Acute	Less than 90°
Right	90°
Obtuse	Between 90° and 180°
Straight	180°

More often than not, you’ll work with multiple angles in a single question. Therefore, it’s worth noting two likely familiar terms that involve working with two or more angles: complementary and supplementary angles. Two angles are complementary if their measures add up to 90°; if their measures add up to 180°, the angles are supplementary.

Note

Two angles need not be adjacent to be complementary or supplementary.

Intersecting lines create angles with special relationships you’ll need to know as well. When two lines intersect, adjacent angles are supplementary, and vertical angles (two angles opposite a vertex) are equal, or congruent. Take a look at the following figure for an example.

The angles marked a° and b° are supplementary; therefore, a + b = 180. The angle marked a° is vertical (and thus equal) to the one marked 60°, so a = 60. With this new information, you can find b: a + b = 60 + b = 180, so b = 120.

Note

Unless otherwise noted, figures on the SAT are drawn to scale. Be especially careful when you see “Note: Figure not drawn to scale.”

When two parallel lines are intersected by another line (called a transversal), all acute angles are equal, and all obtuse angles are equal. Additionally, corresponding angles are angles that are in the same position but on different parallel lines/transversal intersections; they are also equal. Furthermore, alternate interior angles and alternate exterior angles are equal. Alternate interior angles are angles that are positioned between the two parallel lines on opposite sides of the transversal, whereas alternate exterior angles are positioned on the outside of the parallel lines on opposite sides of the transversal. Consider the following figure:

• Line 1 and Line 2 are parallel and cut by transversal ℓ.
• Angles a, d, e, and h are obtuse and equal.
• Angles b, c, f, and g are acute and equal.
• Angle pairs (b and f), (c and g), (a and e), and (d and h) are corresponding angles.
• Angle pairs (a and h) and (b and g) are alternate exterior angles.
• Angle pairs (d and e) and (c and f) are alternate interior angles.

Below is a summary of the essential theorems related to parallel lines that you'll need to know. Notice that the converse of most of these theorems is also true.

Angle Theorem	Definition
Alternate Interior Angles	If two parallel lines are cut by a transversal, the alternate interior angles are congruent. If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
Alternate Exterior Angles	If two parallel lines are cut by a transversal, the alternate exterior angles are congruent. If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel.
Corresponding Angles	If two parallel lines are cut by a transversal, the corresponding angles are congruent. If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
Vertical Angles	If two parallel lines are cut by a transversal, the vertical angles are congruent.

Note

Parallel lines cut by a transversal are powerful: Even if you know only one angle measure, you can get the other seven in mere seconds by knowing these rules. 

Triangles

Lines and angles form the basis of triangles—some of the most commonly occurring shapes on the SAT. Luckily, triangle questions usually don’t involve a lot of complex algebra and are a great way to earn a few quick points on Test Day. Having a good command of triangle properties will help you recognize and solve these questions quickly. Many seemingly difficult questions will become easier once you can confidently speak the language of triangles.

All triangles follow the rules listed here, regardless of the type of triangle, so take the time now to get comfortable with these rules. 

Triangle Theorem	Definition
Triangle Sum and Exterior Angle Theorems	Interior angles add up to 180°. An exterior angle equals the sum of the two opposite interior angles.
Isosceles Triangle Theorems	If two sides of a triangle are congruent, the angles opposite them are congruent. If two angles of a triangle are congruent, the sides opposite them are congruent.
Triangle Inequality Theorem	The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. The difference of the lengths of any two sides of a triangle must be less than the length of the third side.
Side-Angle Relationship	In a triangle, the longest side is across from the largest angle. In a triangle, the largest angle is across from the longest side.
Mid-Segment Theorem	A triangle mid-segment (or midline) is parallel to one side of the triangle and joins the midpoints of the other two sides. The mid-segment’s length is half the length of the side to which it is parallel.

The corresponding angles and side lengths of congruent triangles are equal. Similar triangles have the same angle measurements and proportional sides. In the figure below, ΔABC and ΔDEF have the same angle measurements, so the side lengths can be set up as the following proportion: .

Drawing multiple heights in one triangle creates similar triangles, as shown in the diagram below. When you encounter a question like this, redrawing the similar triangles with their angles and sides in the same positions will help keep information in order.

Complex Figures

Complex figures are also a recurring SAT geometry topic, particularly ones that involve triangles. A complex figure is not a shape such as a dodecahedron. Instead, a complex figure is usually a larger shape that is composed of multiple (familiar) shapes; these can be obvious or cleverly hidden. These figures can always be broken down into squares, rectangles, triangles, and/or circles. No matter how convoluted the figure, following the guidelines here will lead you to the correct answer on Test Day.

• Transfer information from the question stem to the figure. If a figure isn’t provided, draw one!
• Break the figure into familiar shapes.
• Determine how one line segment can play multiple roles in a figure. For example, if a circle and triangle overlap correctly, the circle’s radius might be the triangle’s hypotenuse.
• Work from the shape with the most information to the shape with the least information.

Now use your knowledge of lines, angles, triangles, and complex figures to answer a couple of test-like questions.

1. 

   In the figure above, and are parallel. What is the value of x ?

   1.   60
   2.   70
   3.   80
   4. 110

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking	Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go You’re asked for the value of x.
Step 2: Choose the best strategy to answer the question Look for familiar shapes within the given figure. There are three triangles present. Because and are parallel, BFDA is a trapezoid. Although x might seem far removed from the known angles, you can find it. It will just take more than one step. In ΔFDC, only one angle is missing, so you can solve for it and fill it in easily. Note that angle DFC and angle BFG are vertical angles, so m angle BFG is also 50°. At this point, you have two of the three angles in ΔBFG, so you can solve for the third. angle FBG and angle BAD are corresponding angles (and are therefore congruent). You can now conclude that x = 70.
Step 3: Check that you answered the right question You’re asked for x, so select (B), and you’re done.	x = 70

Let's try another test-like question. Refer back to the properties and theorems as needed.

2. 

   In the figure above, is a mid-segment of ΔPQR. If ST = 4x + 9 and PR = 16x + 6, what is the length of  ?

   1.   0.25
   2.   1.5
   3. 15
   4. 30

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking	Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go You must find the length of
Step 2: Choose the best strategy to answer the question
A triangle’s mid-segment has a length that is half that of the side to which it is parallel. In other words, . You’re given an expression for each segment, so plug in the expressions and solve for x. The question asks for the length of Substitute the value you got for x back into the expression for PR and simplify.
Step 3: Check that you answered the right question The correct choice is (D).

The Pythagorean Theorem, Pythagorean Triplets, & Special Right Triangles

The Pythagorean theorem is one of the most fundamental equations in geometry, and it will be of great use to you on the SAT. Common Pythagorean triplets and special right triangle ratios that originate from this formula will also serve you well on Test Day.

The Pythagorean theorem is an important triangle topic that you are probably familiar with already. If you know the lengths of any two sides of a right triangle, you can use the Pythagorean theorem equation to find the missing side. The equation is expressed as a² + b²= c², where a and b are the shorter sides of the triangle (called legs) and c is the hypotenuse, which is always across from the right angle of the triangle.

Note

The Pythagorean theorem can only be applied to right triangles.

Consider an example: A right triangle has a leg of length 9 and a hypotenuse of length 14. To find the missing leg, plug the known values into the Pythagorean theorem: 9² + b² = 14². This simplifies to 81 + b² = 196, which becomes b² = 115. Take the square root of both sides to get . Because no factors of 115 are perfect squares,   is the answer.

Note

Wait to simplify radicals until you have your final answer. Leave answers in radical form unless a question says otherwise or the answer choices are written as decimals.

Because time is at such a premium on the SAT, time-saving strategies are invaluable, and there are two that will come in handy on triangle questions. The first is knowing common Pythagorean triplets, which are right triangles that happen to have integer sides. These triangles show up very frequently on the SAT. The two most common are 3-4-5 and 5-12-13. Multiples of these (e.g., 6-8-10 and 10-24-26) can also pop up, so watch out for them as well. The beauty of these triplets is that if you see any two sides, you can automatically fill in the third without having to resort to the time-consuming Pythagorean theorem.

The second time-saving strategy involves recognizing special right triangles. Like Pythagorean triplets, special right triangles involve a ratio comparing the lengths of a right triangle’s legs and hypotenuse, but with these triangles, you only need to know the length of one side in order to calculate the other two. These triangles are defined by their angles.

The ratio of the sides of a 45-45-90 triangle is x : x : x, where x is the length of each leg and is the length of the hypotenuse.

The ratio of the sides of a 30-60-90 triangle is x : x : 2x, where x is the shorter leg, is the longer leg, and 2x is the hypotenuse.

While the Pythagorean theorem can always be used to solve right triangle questions, it is not always the most efficient way to proceed. Further, many students make errors when simplifying radicals and exponents. The Pythagorean triplets and special right triangles allow you to save time and avoid those mistakes. Use them whenever possible!

Note

Although you will be given the special right triangle ratios on Test Day, you can save yourself some time by memorizing them. That way you won’t have to keep flipping back to the formula page! You should also commit Pythagorean triplets to memory to bypass the Pythagorean theorem on Test Day.

Area of a Triangle

On Test Day, you might also be asked to find the area of a triangle or use the area of a triangle to find something else. The area of a triangle can be determined using , where b is the triangle base and h is the triangle height.

When you have a right triangle, you can use the legs as the base and the height. If the triangle isn’t a right triangle, you’ll need to draw the height in, as demonstrated in the figure shown. Remember that the height must be perpendicular to the base.

A final note about triangles: You are likely to see triangle questions involving real-world situations. But don’t fret: All you need to do is follow the Kaplan Strategy for Translating English into Math. Extract the geometry information you need, then solve.

Take a look at another test-like triangle question.

3. When Ted earned his driver's license, he wanted his first solo drive to be to a friend’s house. Previously, Ted had always biked to his friend’s house and was able to cut through the yards of neighbors and a park in order to bike there in a straight line. In his car, however, Ted had no choice but to follow the streets. As a result, he traveled 6 miles east, 6 miles south, and 2 more miles east. How much shorter, in miles, is Ted’s bike route than his car route?

   

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking	Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go You’re given the lengths of various sections of streets. If only there were some way to arrange this information visually.
Step 2: Choose the best strategy to answer the question To translate this question into a geometry problem, draw a figure. Sketch the streets, label the distances, and add a direct line between the start and destination. Based on the diagram, Ted’s car route is 6 + 6 + 2 = 14 miles. Two triangles are now visible, but you’ll notice the current information is insufficient because you don’t know exactly where the dashed line intersects the vertical line.
Drawing in extra lines reveals a third triangle, and you already know its dimensions. The new triangle has legs measuring 6 miles and 8 miles. Do you see the Pythagorean triplet? It’s a 6-8-10 triangle, meaning Ted’s bike route (the hypotenuse) is 10 miles.	Car: 14 mi Bike: 10 mi
Step 3: Check that you answered the right question The difference is 14 – 10 = 4 miles. Grid in 4, and you’re done!

Triangle CONGRUENCE Theorems

Many students ask why they need to learn how to do proofs when they take geometry. Although you will likely not need them in college (unless you’re a math or computer science major), there is a fundamental skill that comes with constructing proofs: the ability to construct an argument effectively for a statement or position. This skill is critical in numerous situations and fields—for criminal cases in law, research proposals in science, treatment plans in medicine, and others—so it’s a powerful tool to have in your skill set.

That being said, there’s no question that proofs can be unnerving. The good news: You will not need to construct a complete proof on the SAT. The language of certain questions might still be slightly intimidating, but it will be far more manageable than a full-blown proof.

There are several theorems that can be used to prove two triangles are congruent; these are summarized in the following table. Make sure you are comfortable with all of them—you may need to determine that two triangles are congruent in order to find a side length or an angle measure in one or both of the triangles.

Triangle Congruence Theorem	Notation	Diagram
If three sides of one triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent.	SSS (side-side-side)
If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent.	SAS (side-angle-side)
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent.	AAS (angle-angle-side)
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent.	ASA (angle-side-angle)
If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the two triangles are congruent.	HL (hypotenuse-leg)
An angle or line segment is congruent to itself.	Reflexive Property
Corresponding parts of congruent triangles are congruent.	CPCTC

Note

There are two bogus congruence “theorems” to watch out for: AAA and SSA. Two triangles with identical angles are always similar, but they aren’t necessarily congruent. If you have two triangles with two adjacent congruent sides and a congruent angle outside them, the third side can be two different lengths. SSA is valid only with right triangles, in which case you call it HL. Never use AAA.

Quadrilateral Theorems

Quadrilaterals are four-sided figures, with interior angles that add up to 360°. You will likely not see a question solely about quadrilaterals; if a quadrilateral appears at all, it will likely contain hidden triangles. However, the properties of the quadrilateral will allow you to deduce information about the triangles present, so make sure you know the basic properties of the most common quadrilaterals.

Given the high likelihood of a Test Day geometry problem containing hidden triangles, you should familiarize yourself with the types of quadrilaterals that are most likely to have useful triangles within.

Parallelogram Theorems	Properties
Parallelogram	Both pairs of sides are parallel. Both pairs of sides are congruent. Both pairs of opposite angles are congruent. An angle is supplementary to both angles adjacent to it. The diagonals bisect each other.
Rhombus	It has all the properties of a parallelogram. All four sides are congruent. Diagonals bisect angles and are perpendicular.
Rectangle	It has all the properties of a parallelogram. All angles are right angles. Diagonals are congruent.
Square	It has all the properties of a parallelogram, rhombus, and rectangle. All sides are congruent.

A word of caution: Do not make any assumptions about a test figure that go beyond the information provided in the question. It’s tempting to assume a quadrilateral is a rectangle or other more “friendly” shape, but unless this is proven or stated in the question (or indicated in the figure), don’t do it!

Introduction to Circles

You already know the SAT can ask a variety of questions about lines, angles, and triangles; it can also test you on your knowledge of circles. Keep reading for a refresher on these ubiquitous shapes.

Circle Anatomy & Basic Formulas

There are a number of circle traits you should know for Test Day. 

1. Radius (r): The distance from the center of a circle to its edge
2. Chord: A line segment that connects two points on a circle
3. Diameter (d): A chord that passes through the center of a circle. The diameter is always the longest chord a circle can have and is twice the length of the radius.
4. Circumference (C): The distance around a circle; the equivalent of a polygon’s perimeter. Find this using the formula C = 2πr = πd.
5. Area: The space a circle takes up, just like a polygon. A circle’s area is found by computing A = πr².
6. Every circle contains 360°. You’ll find out more about this fact’s utility shortly.

As the formulas demonstrate, the radius is often the key to unlocking several other components of a circle. Therefore, your first step for many circle questions will be to find the radius.

Circles on the Coordinate Plane and Their Equations

When you have a circle on the coordinate plane, you can describe it with an equation. The equation of a circle in standard form is as follows:

(x − h)² + (y − k)² = r²

In this equation, r is the radius of the circle, and h and k are the x- and y-coordinates of the circle’s center, respectively: (h, k).

Note

A circle is one of the four conic sections, which are made by slicing a double cone with a plane. The parabola is another conic section that you read about in chapter 10. The ellipse and hyperbola (with standard forms and , respectively) will not be tested on the SAT. Watch out for them in trap answer choices, however!

You might also see what is referred to as general form:

x² + y² + Cx + Dy + E = 0

At first glance, the general form probably doesn’t resemble the equation of a circle, but the fact that you have an x² term and a y² term with coefficients of 1 is your indicator that the equation does indeed graph as a circle. To convert to standard form, complete the square for the x terms, then repeat for the y terms. Refer to chapter 10 for a review of completing the square.

Note

The x² and y² terms in the equation of a circle occasionally have coefficients other than 1 (but the coefficients must be equal). If this is the case, simply divide all terms in the equation by the coefficient to eliminate them.

Ready to try a circle question?

4. x² + 6x + y² − 8y = 171

   The equation of a circle in the xy-plane is shown above. What is the positive difference between the x- and y-coordinates of the center of the circle?

   

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking	Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go You need to determine the positive difference between the x- and y-coordinates of the center of the circle.
Step 2: Choose the best strategy to answer the question
You’ll need to rewrite the equation in standard form to find what you need. The coefficients of the x and y terms are even, so consider completing the square for x and y. Divide b (from the x term) by 2 and square the result. Repeat for y. Then add the resulting amounts to both sides of the equation. Factor to write the equation in standard form. The center of the circle is (–3, 4); subtract –3 from 4 to get the positive difference.	center: (–3, 4)
Step 3: Check that you answered the right question The positive difference between the two coordinates is 7.	4 – (–3) = 7

Circle Ratios: Arcs, Central Angles, & Sectors

The SAT can ask you about parts of circles as well. There are three partial components that can be made in a circle: arcs, central angles, and sectors. These circle pieces are frequently used in proportions with their whole counterparts, so the ability to set up ratios and proportions correctly is of utmost importance for these questions.

1. An arc is part of a circle’s circumference. Both chords and radii can cut a circle into arcs. The number of arcs present depends on how many chords and/or radii are present. If only two arcs are present, the smaller arc is called the minor arc, and the larger one is the major arc. If a diameter cuts the circle in half, the two formed arcs are called semicircles. An arc length can never be greater than the circle’s circumference.
2. When radii cut a circle into multiple (but not necessarily equal) pieces, the angle at the center of the circle contained by the radii is the central angle. Because a full circle contains 360°, a central angle measure cannot be greater than this.
3. Radii splitting a circle into pieces can also create sectors, which are parts of the circle’s area. The area of a sector cannot be greater than its circle’s total area.

Here’s a summary of the ratios formed by these three parts and their whole counterparts.

Notice that all of these ratios are equal. Intuitively, this should make sense: When you slice a pizza into four equal slices, each piece should have of the cheese, sauce, crust, and toppings. If you slice a circle into four equal pieces, the same principle applies: Each piece should have of the degrees, circumference, and area.

Inscribed Angle Theorem

An angle whose vertex is on the edge of the circle is called an inscribed angle. As this vertex moves along the edge, the measure of the inscribed angle remains constant as long as the minor arc created (in other words, isolated or subtended by the chords) does not change. When the chords that create an inscribed angle subtend the same minor arc that a pair of radii do, a special relationship appears: The central angle measure is twice that of the inscribed angle.

Arcs Formed Between Parallel Chords

Another theorem states that two parallel chords will intercept two congruent arcs; see the following diagram for an example. The congruent arcs will be between the chords.

Tangent Lines

A tangent line touches a circle at exactly one point and is perpendicular to a circle’s radius at the point of contact. The following diagram demonstrates what this looks like.

The presence of a right angle opens up the opportunity to draw otherwise hidden shapes, so pay special attention to tangents when they’re mentioned. They often come up in complex figure questions.

The next question will give you a chance to see how these properties of circles can be tested on the SAT.

5. An orange with a diameter of 2 inches is sitting on a counter. If the distance from the center of the orange to the edge of the counter is 10 inches, how many inches is it between the point where the orange sits on the counter and the counter edge?

   1. 
   2. 
   3. 
      
   4. 

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking	Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go
You must calculate the distance between where the orange sits to the edge of the counter.
Step 2: Choose the best strategy to answer the question
Draw a cross section of the orange on the counter; you’ll see this is a circle with a tangent line. Draw in a perpendicular radius and a line from the center of the orange to the edge of the counter to reveal a right triangle.
Use the Pythagorean theorem to calculate the distance requested.
Step 3: Check that you answered the right question
Simplify the radical; (B) is correct.

Introduction to 3-D shapes

Over the last several pages, you learned about two-dimensional (2-D) shapes and how to tackle SAT questions involving them. Now you’ll learn how to do the same for questions containing three-dimensional (3-D) shapes, also called solids. There are several different types of solids that might appear on Test Day—rectangular solids, cubes, cylinders, prisms, spheres, cones, pyramids—so it is critical that you be familiar with them.

The following is a diagram showing the basic anatomy of a 3-D shape.

A face (or surface) is a 2-D shape that acts as one of the sides of the solid. Two faces meet at a line segment called an edge, and three faces meet at a single point called a vertex.

Keep reading for more on types of 3-D shapes and questions you could be asked about them.

Volume

Volume is the amount of 3-D space occupied by a solid. This is analogous to the area of a 2-D shape like a triangle or circle. You can find the volume of many 3-D shapes by finding the area of the base and multiplying it by the height. We’ve highlighted the base area components of the formulas in the following table using parentheses.

Rectangular Solid	Cube	Right Cylinder
(l × w) × h	(s × s) × s = s3	(π × r 2) × h

These three 3-D shapes are prisms. Almost all prisms on the SAT are right prisms; that is, all faces are perpendicular to those with which they share edges.

Following are some examples of less commonly seen prisms.

Triangular Prism	Hexagonal Prism	Decagonal Prism

Like the rectangular solids, cubes, and cylinders you saw earlier, these right prisms use the same general volume formula (V = A_base × h).

Note

You might not be told explicitly the area of the base of a prism, in which case you’ll need to rely on your two-dimensional geometry expertise to find it before calculating the volume.

More complicated 3-D shapes include the right pyramid, right cone, and sphere. The vertex of a right pyramid or right cone will always be centered above the middle of the base. Their volume formulas are similar to those of prisms, albeit with different coefficients.

Note

Some of these formulas might look daunting, but don’t fret—you won’t have to memorize them for Test Day. They’ll be provided on the reference page at the beginning of each math section.

Right Rectangular Pyramid	Right Cone	Sphere

Note

A right pyramid can have any polygon as its base; the square variety is the one you’re most likely to see. Also note that the vertex above the base of a right pyramid or cone is not necessarily formed by an intersection of exactly three faces, as in prisms, but it is still a single point and is still called a vertex.

Surface Area

Surface area is the sum of the areas of all faces of a solid. You might liken this to determining the amount of wrapping paper needed to cover all faces of a solid.

To calculate the surface area of a solid, simply find the area of each face using your 2-D geometry skills, then add them all together.

Note

You won’t be expected to know the surface area formulas for right pyramids, right cones, and spheres; they’ll be provided if you need them. However, you could be asked to find the surface area of a prism, in which case you’ll be given enough information to find the area of each surface of the solid.

You might think that finding the surface area of a solid with many sides, such as a 10-sided right octagonal prism, is a tall order. However, you can save time by noticing a vital trait: This prism has two identical octagonal faces and eight identical rectangular faces. Don’t waste time finding the area of each of the 10 sides; find the area of one octagonal face and one rectangular face instead. Once complete, multiply the area of the octagonal face by 2 and the area of the rectangular face by 8, add the products together, and you’re done! The same is true for other 3-D shapes such as rectangular solids (including cubes), other right prisms, and certain pyramids.

If you’re ready to test your knowledge of 3-D shapes, check out the next question.

6. Desiree is making apple juice from concentrate. The cylindrical container of concentrate has a diameter of 7 centimeters and a height of 12 centimeters. To make the juice, the concentrate must be diluted with water so that the mix is 75% water and 25% concentrate. If Desiree wishes to store all of the prepared juice in one cylindrical pitcher that has a diameter of 10 centimeters, what must its minimum height in centimeters be (rounded to the nearest centimeter)?

   

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking	Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go
You need to find the minimum height of a pitcher with a diameter of 10 centimeters that can hold the entire amount of juice after it has been properly mixed.
Step 2: Choose the best strategy to answer the question
Before finding the height of the pitcher, you’ll need to determine the volume of the juice after it has been mixed with the water. Determine the volume of concentrate Desiree has, then multiply this amount by 4 (because the concentrate only makes up 25% of the new juice volume) to calculate the post-dilution volume. Use the volume formula again to find the minimum height of the juice pitcher.
Step 3: Check that you answered the right question
Round to the nearest centimeter, and you’re done!	24

Note

Double-check your rounding on questions like this: Make sure you’ve rounded to the correct place and done so correctly.