Fiveable SAT Math: Breaking Down the Reference Sheet

The SAT Math Reference Sheet

The SAT gives you a reference sheet at the start of each math section. It contains formulas for areas, volumes, special right triangles, and a few key geometric facts. You don't need to memorize these formulas since they're provided, but you do need to know how to use them quickly and correctly. Familiarity with the reference sheet before test day means you won't waste time figuring out what each formula means under pressure.

During the SAT, you'll be given the following reference sheet which consists of essential formulas you need to know, such as the area and circumference of a circle, area of common shapes, volume of solids, special right triangle ratios, and 3 fundamental geometric rules.

Area and Circumference of a Circle

The area of a circle is $A = \pi r^2$, where:

• A is the area (the space inside the circle)
• π ≈ 3.14159
• r is the radius (the distance from the center to the edge)

The circumference of a circle is $C = 2\pi r$ or equivalently $C = \pi d$, where:

• C is the circumference (the distance around the circle)
• r is the radius
• d is the diameter (the full distance across the circle through the center, so $d = 2r$)

💡 Since $d = 2r$, both circumference formulas give you the same answer. Use whichever one matches the information the problem gives you.

Area of a Rectangle

The formula for the area of a rectangle is $A = lw$, where:

• A is the area (the space inside the rectangle)
• l is the length
• w is the width

💡 On the SAT, rectangles sometimes appear as parts of composite shapes. If a problem shows an irregular figure, look for rectangles you can break it into.

Area of a Triangle

The formula for the area of a triangle is $A = \frac{1}{2}bh$, where:

• A is the area
• b is the base (any side of the triangle)
• h is the height, measured perpendicular to the base

💡 The height is not always a side of the triangle. For obtuse triangles, the height may fall outside the triangle itself. Always look for the perpendicular distance from the base to the opposite vertex.

Pythagorean Theorem

The Pythagorean theorem applies only to right triangles. It states:

$a^2 + b^2 = c^2$

Here, c is the hypotenuse (the side opposite the right angle, and always the longest side), while a and b are the two legs.

If you know any two sides of a right triangle, you can find the third:

1. To find the hypotenuse: $c = \sqrt{a^2 + b^2}$
2. To find a leg: $a = \sqrt{c^2 - b^2}$

💡 Watch for common Pythagorean triples on the SAT: 3-4-5, 5-12-13, 8-15-17, and their multiples (like 6-8-10). Recognizing these saves time.

Special Right Triangles

Two types of right triangles have side lengths that follow fixed ratios. Knowing these ratios lets you find missing sides without using the Pythagorean theorem.

45°– 45°– 90° Triangle

The side ratio is $x : x : x\sqrt{2}$

• The two legs are equal in length (both $x$)
• The hypotenuse is $x\sqrt{2}$

So if a leg is 5, the hypotenuse is $5\sqrt{2}$. If the hypotenuse is 10, each leg is $\frac{10}{\sqrt{2}} = 5\sqrt{2}$.

30°– 60°– 90° Triangle

The side ratio is $x : x\sqrt{3} : 2x$

• The shortest side (opposite 30°) is $x$
• The medium side (opposite 60°) is $x\sqrt{3}$
• The hypotenuse (opposite 90°) is $2x$

So if the short leg is 4, the longer leg is $4\sqrt{3}$ and the hypotenuse is 8.

💡 A common mistake is mixing up which leg gets the $\sqrt{3}$. Remember: the longer leg (opposite the bigger acute angle, 60°) gets the $\sqrt{3}$.

Volume of a Rectangular Prism

$V = lwh$

• V is the volume
• l is the length, w is the width, h is the height

This is just length × width × height. Think of it as the area of the base ($l \times w$) multiplied by how tall the box is.

Volume of a Cylinder

$V = \pi r^2 h$

• r is the radius of the circular base
• h is the height of the cylinder

The logic here: $\pi r^2$ gives you the area of the circular base, and multiplying by $h$ extends it upward.

Volume of a Sphere

$V = \frac{4}{3}\pi r^3$

• r is the radius of the sphere

💡 Notice the radius is cubed here, not squared. That's because volume is three-dimensional.

Volume of a Cone

$V = \frac{1}{3}\pi r^2 h$

• r is the radius of the circular base
• h is the height of the cone

💡 A cone is exactly $\frac{1}{3}$ the volume of a cylinder with the same base and height. If you see a problem comparing the two, that relationship is the key.

Volume of a Pyramid

$V = \frac{1}{3}lwh$

• l and w are the length and width of the rectangular base
• h is the height from the base to the apex (tip)

Just like the cone, a pyramid's volume is $\frac{1}{3}$ of the corresponding prism (same base, same height). The pattern: pointy solids are always $\frac{1}{3}$ of their flat-topped counterparts.

Three Geometric Facts on the Reference Sheet

The reference sheet also includes three rules that show up constantly:

• The number of degrees in a circle is 360
• The number of degrees in a triangle is 180
• The number of radians in a circle is $2\pi$

These seem simple, but they're the foundation for many SAT geometry questions. For example, if you're given two angles of a triangle, you can always find the third by subtracting from 180°.

💫 Conclusion

The reference sheet is a tool, not a crutch. You'll use it most effectively if you already understand what each formula does and when to apply it. Practice using these formulas with real SAT problems so that on test day, a quick glance at the sheet is all you need.