# SAT Geometry and Trigonometry: Formulas and Tips | topin

SAT guidesGeometry and Trigonometry

Section

Math, about 15% of questions (5–7 scored)

Skills

Area and volume; lines, angles and triangles; right triangles and trigonometry; circles

Reference sheet

Area and volume formulas, Pythagoras, 30-60-90 and 45-45-90 triangles

Not on the sheet

Circle equation, SOH-CAH-TOA, slope and distance formulas

Figures

Drawn to scale unless the question says otherwise

## How it works

Questions ask for lengths, angles, areas and volumes from a figure or description, use similar triangles and trigonometric ratios, and work with circles both as shapes (arcs, sectors, angles) and as equations in the xy-plane. Harder versions involve completing the square to find a circle's centre and radius, sin x = cos(90° − x), radians, and how area and volume scale.

## Worked example

The equation of a circle in the xy-plane is x² + y² − 6x + 8y = 11\. What is the radius of the circle?

1. A√11
2. B6
3. C11
4. D36
Show the answer

B: 6

Complete the square to get the form (x − h)² + (y − k)² = r². Group the terms: (x² − 6x) + (y² + 8y) = 11\. Add 9 to complete x² − 6x and 16 to complete y² + 8y, on both sides: (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36\. So r² = 36 and r = 6, with the centre at (3, −4). √11 comes from treating 11 as r²; 36 is r², not r.

## A method for Geometry and Trigonometry

1. 1  
### Open the reference sheet early  
Know what is on it so you never waste time recalling a volume formula, and learn what is not on it.
2. 2  
### Mark the figure  
Write given lengths and angles on your scratch paper copy of the figure, and look for right triangles and similar triangles hidden inside it.
3. 3  
### Use similar triangles and ratios  
Similar triangles have equal angles and proportional sides. Lengths scale by k, areas by k² and volumes by k³.
4. 4  
### SOH-CAH-TOA and complementary angles  
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. In a right triangle, the sine of one acute angle equals the cosine of the other.
5. 5  
### Put circle equations in standard form  
(x − h)² + (y − k)² = r² gives the centre (h, k) and radius r. Complete the square when the equation is expanded.

## Traps to avoid

* Reporting r² as the radius, or the centre with the wrong signs.
* Doubling the volume when every dimension doubles, instead of multiplying it by 8.
* Using degrees when the question gives or asks for radians.
* Assuming a figure is not to scale on the SAT, when figures are drawn to scale unless stated otherwise.

## Geometry and Trigonometry FAQs

What formulas are given on the SAT?

Area of a circle, rectangle and triangle; circumference; the Pythagorean theorem; the 30-60-90 and 45-45-90 triangles; and volumes of a rectangular prism, cylinder, sphere, cone and pyramid. It also notes that a circle has 360° or 2π radians and a triangle's angles sum to 180°.

Is trigonometry on the SAT?

Yes, at a basic level: sine, cosine and tangent in right triangles, the relationship sin x = cos(90° − x), and converting between degrees and radians.

Are SAT figures drawn to scale?

Yes, unless the question says otherwise, so a figure can help you estimate and check an answer.

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