For SAT geometry and trigonometry, begin with a labeled sketch and the quantity requested. Identify whether you need a length, angle, area, volume, or ratio. Then choose a relationship that connects the information given to that quantity. A formula is useful only when you understand what its variables represent.
Geometry and Trigonometry is one of the official Math domains, and Bluebook provides a reference sheet. This guide’s worked examples are original. Review the Math overview and Bluebook’s testing tools.
Build a diagram that records conditions
If a question describes a right triangle, mark the right angle. If two lines are parallel, label that relationship. If a point is the center of a circle, distinguish it from a point on the circumference.
Do not treat a drawing’s appearance as a measurement. A line may look like a diameter without being identified as one, and a triangle may look isosceles without two equal sides being established.
List the given information separately from what you infer. For example:
- Given: triangle ABC is right at B.
- Given: AB = 6 and BC = 8.
- Derived: AC = 10 by the Pythagorean theorem.
This separation makes it easier to notice when an assumption slipped into your solution.
Use angle relationships before side calculations
The angles in a triangle sum to 180 degrees. If two measure 48 and 67 degrees, the third is 65 degrees.
An exterior angle formed by extending one side equals the sum of the two nonadjacent interior angles. For those same 48- and 67-degree angles, the corresponding exterior angle is 115 degrees.
Vertical angles are equal. Adjacent angles on a straight line sum to 180 degrees. When parallel lines are cut by a transversal, corresponding and alternate interior angle relationships can provide equal angles.
The parallel condition matters. Do not apply a parallel-line angle rule just because two drawn lines appear to run in similar directions.
For a problem with several variables, use a short equation rather than mental subtraction through a complicated diagram. If two supplementary angles are 3x + 10 and 2x + 20, then 5x + 30 = 180, giving x = 30.
Connect right-triangle sides correctly
The Pythagorean theorem is a² + b² = c², where c is the hypotenuse opposite the right angle.
For legs six and eight, c² = 36 + 64 = 100, so c = 10.
If the hypotenuse is thirteen and one leg is five, the other leg satisfies b² = 169 − 25 = 144, so b = 12. Adding the squares of thirteen and five would incorrectly treat both as legs.
Recognizable triples can save work, but use them as checked relationships rather than guesses. A triangle with sides 9, 12, and 15 is a scaled 3-4-5 triangle because all three lengths share the same scale factor.
If the side lengths are expressed with radicals, keep exact values until the question asks for rounding.
Use similarity to establish a scale factor
Similar triangles have equal corresponding angles and proportional corresponding sides.
Suppose a small triangle has sides 5, 7, and 8, and a similar triangle’s side corresponding to 5 measures 15. The scale factor from small to large is three. The other sides are therefore 21 and 24.
Pair corresponding sides before writing a proportion. A correct arithmetic operation with mismatched sides produces an incorrect scale factor.
Lengths scale by k, areas by k², and volumes of similar solids by k³. If two similar triangles have a side ratio of 2:3, their area ratio is 4:9.
For an original example, a model’s linear dimensions are one-fourth of the full object’s dimensions. Its volume is one sixty-fourth of the full object’s volume, not one-fourth. Three dimensions each contribute a factor of one-fourth.
Choose the correct base and height
Triangle area is one-half times base times perpendicular height. A slanted side is not automatically the height.
For a parallelogram with a base of twelve and perpendicular height of five, the area is sixty square units, regardless of the slanted side’s length.
For a composite figure, divide the shape into parts with known area formulas. If a rectangle has area eighty and a triangular cutout has base four and height six, subtract twelve to obtain sixty-eight.
Label units in the final answer. A perimeter is measured in linear units, while an area is measured in square units. If an answer’s units do not match the requested quantity, recheck which formula you used.
A surface-area question and a volume question may use the same dimensions but count different things. Sketch the faces or layers before combining them.
Distinguish circle radius, diameter, and arc length
The diameter is twice the radius. Circumference is 2πr, and area is πr².
For a circle with diameter fourteen, the radius is seven. Its circumference is 14π and its area is 49π.
A sector with central angle 90 degrees is one-quarter of a full circle. In a circle of radius eight, its arc length is one-quarter of 16π, or 4π. Its area is one-quarter of 64π, or 16π.
Arc length and sector area use the same fraction of the circle but multiply different whole-circle quantities. Read whether the question requests a boundary distance or an enclosed region.
For a central angle measured in radians, arc length is rθ. A radius of six and angle π/3 produce an arc length of 2π. Do not substitute a degree measure directly into this radian formula.
Recognize a circle’s equation
In the coordinate plane, a circle with center (h, k) and radius r has equation:
(x − h)² + (y − k)² = r²
For (x + 2)² + (y − 5)² = 36, the center is (−2, 5) and the radius is six.
The signs inside the parentheses are easy to reverse. Find the values that make both squared terms zero; those values identify the center.
An expanded equation may require completing the square. From x² + y² − 6x + 4y = 12, regroup and complete:
(x − 3)² − 9 + (y + 2)² − 4 = 12
Thus (x − 3)² + (y + 2)² = 25, with center (3, −2) and radius five. Remember that adding a square-completion term changes the constant as well.
Define trigonometric ratios relative to the chosen angle
For an acute angle θ in a right triangle:
- sin θ = opposite / hypotenuse.
- cos θ = adjacent / hypotenuse.
- tan θ = opposite / adjacent.
“Opposite” and “adjacent” depend on which acute angle you choose. The hypotenuse does not change.
In a 5-12-13 triangle, let θ be opposite the side of length five. Then sin θ = 5/13, cos θ = 12/13, and tan θ = 5/12.
For the other acute angle, the opposite and adjacent legs switch. This explains the complementary relationship sin θ = cos(90° − θ).
If the question gives a sine ratio and asks for another side ratio, draw a representative triangle. A ratio describes proportional lengths; it does not require the actual sides to be those exact integers.
Check calculator angle mode
A trigonometric calculation depends on whether angles are measured in degrees or radians. Before evaluating an expression involving a specified angle, check that the calculator mode matches it.
You do not need to press a trigonometric button for every problem. Special triangles, complementary angles, and ratio definitions can sometimes provide exact answers more directly.
For example, a 45-45-90 triangle has equal legs and a hypotenuse √2 times a leg. If a leg is three, the hypotenuse is 3√2. An approximate decimal is less informative if the choices use exact radicals.
Practice these decisions in the calculator environment you expect to use, and check the current calculator policy separately when bringing a handheld device.
Review geometry by the missed relationship
Group errors as “wrong corresponding sides,” “radius versus diameter,” “height not perpendicular,” “area versus perimeter,” or “angle mode.” Each label suggests a specific repair.
For your next practice session, solve one similar-triangle problem, one circle problem, and one right-triangle ratio problem. Draw the conditions before choosing formulas, then check units and magnitude.
Use the SAT score calculator for the appropriate practice format, while using your geometry work to identify skills needing attention. SHSPrep’s SAT program includes lessons, targeted practice, and two original full-length mock tests with raw performance results.