**Unit 6: Geometry**
*The SAT Mathematics section contains many questions devoted to testing
geometric concepts as well. A clear understanding of these concepts, as
well as shortcuts that can be used to more quickly solve a geometric
problem, will vastly increase your math score. Work through the topics
below to ensure that you have mastered each concept, using the practice
problems to test your knowledge.*

**Unit 6 Time Advisory**

This unit should take approximately 4 hours to complete.

☐ Subunit 6.1: 1.25 hours

☐ Subunit 6.2: 1 hour

☐ Subunit 6.3: 0.25 hours

☐ Subunit 6.4: 0.25 hours

☐ Subunit 6.5: 0.5 hours

☐ Subunit 6.6: 0.75 hours

**Unit6 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:
- Use relationships between lines and angles to identify the value of
an unknown angle.
- Find the value of unknown sides or angles in a triangle.
- Find the value of an unknown line or angle within a circle.
- Find the value of unknown sides or angles in other polygons.
- Find perimeter, area, and volume of a geometrical shape.

**6.1 Angles and Lines**
- **Reading: ErikTheRed.com: Erik Jacobsen’s “SAT Math Facts and
Formulas”**
Link: ErikTheRed.com: Erik Jacobsen’s “SAT Math Facts and
Formulas” (PDF)

```
Instructions: Please click on the link above and select the link to
the “SAT Math Facts and Formulas” PDF. Read the sections entitled
“Lines,” “Triangles,” “Circles,” “Rectangles and Friends,” and
“Solids” (pages 4-6). Make notes or flashcards of any concepts you
need to review in order to remember for the test. This resource
will cover the topics outlined in subunits 6.1-6.5, as well as any
inclusive sub-subunits.
Reading these sections and taking notes should take approximately 1
hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**6.1.1 Parallel and Perpendicular Lines**
*Note: This topic is covered by the “SAT Math Facts and Formulas”
reading assigned below subunit 6.1. In particular, review the formula
for slope, and the formulas for writing an equation in ‘point-slope’ and
‘slope-intercept’ form, as described under the “Lines (Linear
Functions)” heading.*

**6.1.2 Corresponding, Complementary, and Supplementary Angles**
*Note: This topic is covered by the reading assigned below subunit 6.1.
Corresponding angles are angles with the same degree measure.
Complementary angles are two angles that add up to 90 degrees.
Supplementary angles are two angles that add up to 180 degrees.*

**Web Media: Khan Academy’s “Finding Missing Angles”**Link: Khan Academy’s “Finding Missing Angles” (YouTube)Instructions: The basics of finding missing angles are pretty straightforward. Please click on the link above and watch this short video to brush up. Watch this video as many times as needed to understand how to find missing angles.

Watching this video and pausing to take notes should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**6.2 Triangles**
**6.2.1 Right Triangles**
*Note: This topic is covered by the reading assigned below subunit 6.1.
Make sure you have a good handle on the ‘Special Right Triangles’
(mentioned on the worksheet and reviewed in videos below) and the
Pythagorean triples (explained below). Knowing these values off the top
of your head will help a lot with shortcuts and speed when you encounter
right triangles on the SAT.*

**Web Media: Khan Academy’s “The Pythagorean Theorem”**Link: Khan Academy’s “The Pythagorean Theorem” (YouTube)Instructions: Please click on the link above, and view this video for a brush-up on how to use the Pythagorean Theorem to find the length of a missing side on a right triangle. Simplifying square roots is also covered – understanding how to simplify roots which are not perfect squares is important in order to select the correct answer on the SAT, since the answers are designed to be found without a calculator (and so will be written as simplified roots, rather than decimals).

Watching this video and pausing to take notes should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Reading: Math Open Reference’s “Pythagorean Triples”**Link: Math Open Reference’s “Pythagorean Triples” (HTML)Instructions: This page explains and lists the basic Pythagorean triples – the lowest values that fulfill the Pythagorean Theorem (and will be found as sides of right triangles) when all sides are whole numbers. Become familiar with the Pythagorean triples and their multiples so that you can quickly solve simple right triangle problems, and recognize right triangles by the values of the sides.

Reading this webpage should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Web Media: Khan Academy’s “45-45-90 Triangles” and “30-60-90 Triangles II”**Link: Khan Academy’s “45-45-90 Triangles” (YouTube) and “30-60-90 Triangles II” (YouTube)Instructions: Watch these two videos for a review of the “Special Triangles” and examples of how to work these types of problems.

Watching these videos and pausing to take notes should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**6.2.2 Equilateral Triangles**
*Note: This topic is covered by the “SAT Math Facts and Formulas”
reading assigned below subunit 6.1. In particular, review the
description of an equilateral triangle, as described under the
“Triangles” heading.*

**6.2.3 Isosceles Triangles**
*Note: This topic is covered by the “SAT Math Facts and Formulas”
reading assigned below subunit 6.1. In particular, review the
description of an isosceles triangle, as described under the “Triangles”
heading.*

**6.2.4 Congruent vs. Similar**
*Note: This topic is covered below subunit 6.1. Congruent triangles are
basically identical. They have the same angle measures and are the same
size (the lengths of their sides are the same). Similar triangles have
the same angle measures, but may be different sizes. The lengths of
their sides may be different, but they are proportionately the same
(e.g. if triangle A’s hypotenuse is 3, then a twice as large similar
triangle would have a hypotenuse of 6).*

**6.3 Circles**
**6.3.1 Circumference and Diameter**
*Note: This topic is covered by the “SAT Math Facts and Formulas”
reading assigned below subunit 6.1. In particular, review the formulas
under the “Circles” heading. Remember that diameter (*D*) is the length
of a straight line passing from one point on a circle, through the
midpoint, to a point on the opposite side of the circle. It is
therefore twice the length of the circle’s radius (*r*). The
circumference of a circle is equal to *πD*.*

**6.3.2 Midpoint and Angles**
*Note: This topic is covered by the “SAT Math Facts and Formulas”
reading assigned below subunit 6.1. In particular, review the formulas
for the length of an arc and area of a sector under the “Circles”
heading. Try drawing a circle, and choosing some values for the arc or
angle or sector area, leaving one unknown you can practice finding by
using the appropriate equation.*

**6.3.3 Chords and Tangents**
- **Web Media: Khan Academy’s “Perpendicular Radius Bisects Chord” and
“Right Triangles Inscribed in Circles”**
Link: Khan Academy’s “Perpendicular Radius Bisects
Chord” (YouTube)
and “Right Triangles Inscribed in
Circles”
(YouTube)

```
Instructions: The SAT will often ask a question or two about
finding distances or angles of chords or shapes inscribed inside a
circle. Please click on the links above and watch these videos,
which provide important information that can help you solve such
problems. Be sure to take notes about any rules proven here that
can help you with SAT problems.
Watching these videos should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**6.4 Polygons**
- **Web Media: Khan Academy’s “Sum of Interior Angles of a Polygon”
and “Sum of Exterior Angles of a Convex Polygon”**
Link: Khan Academy’s “Sum of Interior Angles of a
Polygon” (YouTube)
and “Sum of Exterior Angles of a Convex
Polygon”
(YouTube)

```
Instructions: The SAT also likes to ask questions that require you
to understand how to find interior or exterior angles of a polygon.
Please click on the links above and watch these videos to find out
how. Be sure to take notes so that you can review this information
later.
Watching these videos should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**6.5 Perimeter, Area, and Volume**
*Note: List the formulas from “SAT Math Facts and Formulas” (under
subunit 6.1) to find area and volume for each shape. Review and
practice these formulas. The more quickly you can remember them on test
day, the more quickly you can complete the problems.*

**Web Media: Khan Academy’s “Finding the Area and Perimeter of a Non-Standard Polygon”**Link: Khan Academy’s “Finding the Area and Perimeter of a Non-Standard Polygon” (YouTube)Instructions: Please click on the link above and watch this video to learn how to find the area of a polygon without its own specific area formula.

Watching this video and pausing to take notes should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Web Media: Khan Academy’s “Solid Geometry Volume” and “Cylinder Volume and Surface Area”**Link: Khan Academy’s “Solid Geometry Volume” (YouTube) and “Cylinder Volume and Surface Area” (YouTube)Instructions: Please watch these videos for a quick tutorial review of the use of the volume equations found on the “SAT Math Facts and Formulas” PDF (subunit 6.1). The first video includes finding the volume of a cube (starting at 2:30) and the second video covers finding the volume of a cylinder.

Watching these videos and pausing to take notes should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**6.6 Geometry Practice Problems**
- **Assignment: cK-12: Jason Shah’s “SAT Prep - Math Questions” and
“SAT Prep - Math Questions with Explanations”**
Link: cK-12: Jason Shah’s “SAT Prep - Math
Questions” (HTML) and
“SAT Prep - Math Questions with
Explanations” (HTML)

```
Instructions: Please click on the first link above for “Math
Questions” and complete questions 6-8 to apply your new knowledge
and to practice geometry questions.
Once you have completed the questions, please click on the second
link above for “Math Questions with Explanations” and review the
correct answers for questions 6–8. For any questions you missed,
read and review the provided explanations until the correct answer
makes sense to you. You may want to look back at the previous
sections in this unit for help. Once you have studied the correct
answer, think about how you will approach a similar problem in the
future and will avoid making a similar mistake.
Completing this assignment should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**Assignment: cK-12: Jason Shah’s “SAT Prep II - Math Questions” and “SAT Prep II - Math Questions with Explanations”**Link: cK-12: Jason Shah’s “SAT Prep II - Math Questions” (HTML) and “SAT Prep II - Math Questions with Explanations” (HTML)Instructions: Please click on the first link above for “Math Questions” and complete questions 6-8 to apply your new knowledge and to practice geometry questions.

Once you have completed the questions, please click on the second link above for “Math Questions with Explanations” and review the correct answers for questions 6-8. For any questions you missed, read and review the provided explanations until the correct answer makes sense to you. You may want to look back at the previous sections in this unit for help. Once you have studied the correct answer, think about how you will approach a similar problem in the future and will avoid making a similar mistake.

Completing this assignment should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Reading: cK-12: Jason Shah’s “SAT Prep III - Math Questions” and “SAT Prep III - Math Questions with Explanations”**Link: cK-12: Jason Shah’s “SAT Prep III - Math Questions” (HTML) and “SAT Prep III - Math Questions with Explanations” (HTML)Instructions: Please click on the first link above for “Math Questions” and complete questions 6 and 8 to apply your new knowledge and to practice geometry questions.

Once you have completed the questions, please click on the second link above for “Math Questions with Explanations” and review the correct answers for questions 6 and 8. For any questions you missed, read and review the provided explanations until the correct answer makes sense to you. You may want to look back at the previous sections in this unit for help. Once you have studied the correct answer, think about how you will approach a similar problem in the future, and avoid making a similar mistake.

Completing this assignment should take less than 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.