SAT Math: Additional Topics in Math 📐

SAT Math Overview

This guide covers Additional Topics in Math, the fourth and final SAT Math category. It accounts for 6 out of 58 math questions (about 10%), split evenly between the no-calculator and calculator sections. These can be multiple-choice or grid-in questions.

The topics here draw from geometry and trigonometry, so you'll want to brush up on those skills.

Main "Additional Topics in Math" Topic Areas

Here's what falls under this category:

📐 Geometry

• These questions test your understanding of lines, angles, triangles, circles, and other shapes. You may need to solve for area or volume, and the SAT reference sheet provides many of the formulas you'll need.

📈 Coordinate Geometry

• This is about applying geometry concepts on a coordinate plane. The main focus is graphing circles and working with the circle equation.

📏 Trigonometry and Radians

• SOHCAHTOA makes its return here. You'll need to know what radians are and how to compute sine, cosine, and tangent.

🟰 Complex Numbers

• These questions involve arithmetic with complex numbers (numbers that include $i = \sqrt{-1}$). Once the rules click, they're very manageable.

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📐 Geometry

🧠 What You Need to Know: Geometry

Area, Surface Area, and Volume

You should be able to solve for the area, surface area, or volume of a given figure. Most formulas you'll need are on the SAT reference sheet. Check out this breakdown of what's included.

Lines, Midpoints, and Angles

• Lines are straight, one-dimensional figures extending infinitely in both directions. They're defined by two points.
• The midpoint of a line segment is the point that divides it into two equal halves, equidistant from both endpoints. The midpoint formula is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Angles form when two rays share a common endpoint called the vertex. They're measured in degrees, and a full rotation around a point is 360°.
• Common angle types: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), and straight (exactly 180°).
• Vertical angles are the pair of opposite angles formed when two lines intersect. They're always congruent (equal in measure).

Parallel Lines and Transversals

When a transversal (a line crossing two parallel lines) cuts through parallel lines, it creates several angle relationships:

• Corresponding angles: congruent angles in the same position on each parallel line. Example: angles 2 and 6 below.
• Alternate interior angles: congruent angles between the parallel lines on opposite sides of the transversal. Example: angles 3 and 6.
• Alternate exterior angles: congruent angles outside the parallel lines on opposite sides of the transversal. Example: angles 1 and 8.
• Co-interior (same-side interior) angles: angles between the parallel lines on the same side of the transversal. These are supplementary (add up to 180°). Example: angles 3 and 5.

Image Courtesy of Dummies

Perpendicular Lines

Perpendicular lines intersect at a right angle (90°), forming four right angles at the point of intersection.

Right Triangles

You should be comfortable using trigonometric ratios and the Pythagorean theorem to solve right triangle problems.

• The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (the side opposite the right angle) and $a$ and $b$ are the other two sides.
• Know the two special right triangles: 30°-60°-90° and 45°-45°-90°. Their side ratios are given on the reference sheet. In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. In a 45-45-90 triangle, the sides are $1 : 1 : \sqrt{2}$. More details here.

Congruent vs. Similar Triangles

• Congruent triangles have the same shape and size. All corresponding sides and angles are equal.
• Similar triangles have the same shape but different sizes. Corresponding angles are equal, and corresponding sides are proportional.

Image Courtesy of Math Monks

Triangle Inequality Theorem

For any triangle, the length of any side must be less than the sum of the other two sides and greater than the difference of the other two sides. For example, if two sides of a triangle are 5 and 8, the third side must be greater than 3 and less than 13.

Other Shapes

Be familiar with properties of quadrilaterals (squares, rectangles, parallelograms) and regular polygons.

Circles

Know these parts of a circle and how to measure them:

• The radius ($r$) is the distance from the center to any point on the circle. The diameter ($d$) is the distance across the circle through the center: $d = 2r$.
• The circumference is the perimeter of the circle: $C = 2\pi r$.
• A central angle has its vertex at the center of the circle. Its measure equals the measure of the arc it intercepts.
• An inscribed angle has its vertex on the circle. Its measure is half the intercepted arc.
• Arc length is the distance along the curved edge between two points: $L = \frac{\theta}{360} \times 2\pi r$, where $\theta$ is the central angle in degrees.

Image Courtesy of The Geometry Center

Sectors, Tangents, Secants, and Chords

• A sector is the "pie slice" region enclosed by an arc and two radii. Its area is $A = \frac{\theta}{360} \times \pi r^2$.
• A tangent is a line that touches the circle at exactly one point (the point of tangency). A tangent is always perpendicular to the radius at that point.
• A secant is a line that intersects a circle at two points.
• A chord is a line segment connecting two points on the circle. The diameter is the longest possible chord.

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Geometry Notation

Be familiar with standard notation for points, lines, angles, and segments.

Image Courtesy of Ramanathan

🤓 Applying Your Knowledge: Geometry

Area of a Region; Circle Practice

Question and image courtesy of College Board. All credit to College Board.

This question combines several skills, so break it into steps:

1. Figure out what you're solving for. You need the area of the shaded region.
2. Make a plan. The shaded region equals the area of triangle OBC minus the area of the sector bounded by radii OA and OB.
3. Find the sides of triangle OBC. Line OB is 6, and since OA is also a radius, $OA = OB = 6$. The problem states $AC = 6$, so the hypotenuse $OC = OA + AC = 12$. Since the hypotenuse is twice one leg, this is a 30°-60°-90° triangle. The remaining side $BC = 6\sqrt{3}$.
4. Calculate the area of triangle OBC. Using $A = \frac{1}{2}bh$: $A = \frac{1}{2}(6)(6\sqrt{3}) = 18\sqrt{3}$.
5. Calculate the area of the sector. The central angle is 60°, which is $\frac{1}{6}$ of a full circle. The circle's area is $\pi(6)^2 = 36\pi$, so the sector area is $\frac{36\pi}{6} = 6\pi$.
6. Subtract. The shaded region is $18\sqrt{3} - 6\pi$. The answer is choice B.

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📈 Coordinate Geometry

🧠 What You Need to Know: Coordinate Geometry

Coordinate geometry applies your geometry skills on the xy-plane. The main focus here is circles.

The standard equation of a circle is:

$(x - h)^2 + (y - k)^2 = r^2$

where $(h, k)$ is the center and $r$ is the radius.

You should be able to:

• Write the equation of a circle given its graph on a coordinate plane, or sketch the circle given its equation.
• Manipulate a circle equation into standard form. Sometimes you'll be given an expanded equation and need to complete the square to rewrite it.

🤓 Applying Your Knowledge: Coordinate Geometry

Equation of a Circle Practice

Here's an example SAT question (all credit to College Board):

$x^2 + (y + 1)^2 = 4$

The graph of the given equation in the xy-plane is a circle. If the center of this circle is translated 1 unit up and the radius is increased by 1, which of the following is an equation of the resulting circle?

A) $x^2 + y^2 = 5$

B) $x^2 + y^2 = 9$

C) $x^2 + (y + 2)^2 = 5$

D) $x^2 + (y + 2)^2 = 9$

Explanation:

1. Read the original equation. It's already in standard form. The center is at $(0, -1)$ (remember, $(y + 1)$ means $k = -1$), and the radius is $\sqrt{4} = 2$.
2. Translate the center 1 unit up: the y-coordinate goes from $-1$ to $0$. New center: $(0, 0)$.
3. Increase the radius by 1: $2 + 1 = 3$. New radius: 3.
4. Plug into the standard form: $x^2 + y^2 = 9$. The answer is B.

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📏 Trigonometry and Radians

This section tests your understanding of right triangle trigonometry and radian measure.

🧠 What You Need to Know: Trigonometry and Radians

What is a Radian?

A radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle. A full circle is $2\pi$ radians (which equals 360°).

Image Courtesy of Mathematics Monster

Converting Between Degrees and Radians

• Degrees → Radians: Multiply by $\frac{\pi}{180}$. For example, $90° \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.
• Radians → Degrees: Multiply by $\frac{180}{\pi}$. For example, $\frac{\pi}{2} \times \frac{180}{\pi} = 90°$.

Benchmark Trig Values

You should memorize these values for the most commonly tested angles:

The Three Basic Trig Ratios (SOHCAHTOA)

These ratios apply to right triangles:

• Sine: $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine: $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent: $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$

The mnemonic SOHCAHTOA pieces these together: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

One useful relationship: $\sin\theta = \cos(90° - \theta)$. This means the sine of any angle equals the cosine of its complement. The SAT has tested this directly.

Image Courtesy of Catherine S.

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🟰 Complex Numbers

🧠 What You Need to Know: Complex Numbers

A complex number has the form $a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$. The key fact you need is that $i^2 = -1$.

Adding and Subtracting: Combine real parts with real parts and imaginary parts with imaginary parts.

• $(3 + 2i) + (1 + 4i) = 4 + 6i$
• $(5 + 3i) - (2 + i) = 3 + 2i$

Multiplying: Use the distributive property (FOIL) and replace $i^2$ with $-1$.

• $(2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i^2 = 2 + 11i + 12(-1) = -10 + 11i$

Dividing: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $a + bi$ is $a - bi$. This eliminates $i$ from the denominator because $(a + bi)(a - bi) = a^2 + b^2$.

The result of any arithmetic with complex numbers is always another complex number.

🤓 Applying Your Knowledge: Complex Numbers

Dividing Complex Numbers Practice

Here's a question involving division of complex numbers:

Image Courtesy of College Board; All credit to College Board.

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⭐️ Closing

That wraps up Additional Topics in Math and the entire SAT Math guide series. The best way to lock in these concepts is through practice problems. Focus especially on circle equations, special right triangles, and trig ratios, as these appear most frequently.

Good luck studying. 🍀