FREE SAT STUDY GUIDE
SAT Geometry and Trigonometry
Learn triangles, circles, area, volume, and trigonometry through free SAT lessons and worked examples.
These are BuildSAT teaching examples. Use the study workspace to apply the concepts to official practice questions and review your progress.
Open interactive lessons →Lines, angles, and triangles
Lines, angles, and triangles
Use angle relationships and triangle sums.
Angle facts
Vertical angles are equal. A linear pair sums to 180°. Angles around a point sum to 360°. Interior angles of a triangle sum to 180°.
Parallel lines
With parallel lines cut by a transversal, corresponding and alternate interior angles are equal; same-side interior angles are supplementary. The parallel condition matters.
Angle relationships come from the structure of intersecting lines
A straight angle measures 180°. When a ray divides that straight angle, the two adjacent angles add to 180°. Vertical angles are equal because each is supplementary to the same adjacent angle. These relationships allow you to infer unknown measures without measuring the drawing.
Parallelism and triangle sums create additional equations
Parallel lines cut by a transversal generate equal corresponding or alternate interior angles. Without parallelism, those equalities are not guaranteed. In a triangle, the interior angles sum to 180°; an exterior angle equals the sum of the two nonadjacent interior angles because it is supplementary to the third.
Worked example
Two angles of a triangle measure 48° and 67°. Find the third.
- Add known angles: 48 + 67 = 115.
- Subtract from 180.
65°
Watch for this: Do not infer parallel lines or equal lengths from the appearance of a diagram.
Check your understanding
Two angles form a straight line. One is 124°. What is the other?
- 56°
- 66°
- 124°
- 236°
Show answer and explanation
Answer: A. A linear pair sums to 180°, so the other angle is 180 − 124 = 56°.
Lines, angles, and triangles
Similar triangles and corresponding sides
Match corresponding parts before setting up a proportion.
Similarity
Similar triangles have equal corresponding angles and proportional corresponding sides. Two matching angles are enough to establish triangle similarity.
Congruence and scale
Congruent figures have the same size and shape. Similar figures can differ in size. Match sides by their opposite angles or the stated vertex order.
Similar shapes preserve angles while scaling lengths
A dilation multiplies every length by the same positive factor without changing angles. Similar triangles therefore have equal corresponding angles and a constant ratio of corresponding sides. Knowing two matching angles establishes the third because each triangle’s angles sum to 180°.
Correspondence matters more than orientation
A triangle may be rotated or reflected, so the side that looks leftmost is not necessarily the matching side. Match vertices by angle measures or the stated naming order. A side between two named vertices corresponds to the side between their matching vertices in the other triangle.
Worked example
Similar triangles have corresponding shortest sides 3 and 9. The smaller triangle’s longest side is 5. Find the larger longest side.
- The length scale factor is 9/3 = 3.
- Multiply the corresponding length by 3.
15
Watch for this: Keep the numerator and denominator in the same small-to-large or large-to-small order.
Check your understanding
Two similar triangles have a scale factor of 2 from small to large. A small side is 7. What is the matching large side?
- 3.5
- 9
- 49
- 14
Show answer and explanation
Answer: D. Multiply the corresponding length by 2: 7 × 2 = 14.
Right triangles and trigonometry
The Pythagorean theorem and distance
Identify the hypotenuse and solve for a missing length.
Right triangles only
For a right triangle, a² + b² = c², where c is the hypotenuse opposite the right angle. If a leg is missing, subtract before taking a square root.
Coordinate distance
Between (x₁, y₁) and (x₂, y₂), distance = √((x₂ − x₁)² + (y₂ − y₁)²). The horizontal and vertical differences form perpendicular legs.
The theorem relates squares built on a right triangle
For perpendicular legs a and b, the squared hypotenuse is a² + b². The hypotenuse lies opposite the right angle and is the longest side. If the unknown is a leg, subtract the known leg’s square from the hypotenuse’s square. Adding is correct only when finding the hypotenuse’s square.
Distance on a coordinate plane forms a right triangle
The horizontal change Δx and vertical change Δy are perpendicular. Their squares add to the straight-line distance squared. Squaring makes either subtraction order work as long as each coordinate difference uses the correct pair. A distance is nonnegative, so take the positive root.
Worked example
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
- b² = 13² − 5² = 169 − 25 = 144.
- Take the positive root because a length is positive.
12
Watch for this: The longest side is the hypotenuse. A drawing alone does not establish a right angle.
Check your understanding
A right triangle has legs 6 and 8. What is its hypotenuse?
- 14
- 10
- 2
- 100
Show answer and explanation
Answer: B. √(6² + 8²) = √100 = 10.
Right triangles and trigonometry
Special right triangles
Use exact ratios for 45–45–90 and 30–60–90 triangles.
45–45–90
The side ratio is 1 : 1 : √2. Equal legs have length s and the hypotenuse is s√2.
30–60–90
The side ratio opposite 30°, 60°, and 90° is 1 : √3 : 2. Start with the short leg, then scale all three sides.
The 45–45–90 ratio follows from equal legs
Equal acute angles have equal opposite sides. If both legs have length s, the Pythagorean theorem gives c² = s² + s² = 2s², so c = s√2. This derives the 1:1:√2 ratio rather than requiring an unexplained memorized formula.
The 30–60–90 ratio comes from halving an equilateral triangle
An altitude splits an equilateral triangle into two right triangles and bisects its base. If the original side length is 2s, the short leg is s and the hypotenuse is 2s. The remaining leg satisfies b² = 4s² − s² = 3s², so b = s√3.
Worked example
A 30–60–90 triangle has hypotenuse 18. Find the short leg and long leg.
- The hypotenuse is twice the short leg, so the short leg is 9.
- The long leg is √3 times the short leg.
Short leg: 9; long leg: 9√3.
Watch for this: For a 30–60–90 triangle, the side opposite 30° is the shortest.
Check your understanding
A 45–45–90 triangle has legs of length 4. What is the hypotenuse?
- 8
- 4
- 4√2
- 2√2
Show answer and explanation
Answer: C. Use the 1 : 1 : √2 ratio: the hypotenuse is 4√2.
Right triangles and trigonometry
Sine, cosine, tangent, and complementary angles
Choose a ratio using the named angle and known sides.
SOH–CAH–TOA
In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. Opposite and adjacent change when you change the reference angle.
Complementary angles
For acute angles, sin θ = cos(90° − θ). Set the calculator to degrees for degree measures and radians for radian measures.
Trigonometric ratios depend on the angle, not the triangle’s size
Right triangles with the same acute angle are similar, so corresponding side ratios remain constant as the triangle scales. Those ratios define sine, cosine, and tangent. Label opposite and adjacent relative to the requested angle; the hypotenuse always remains opposite the right angle.
Choose the ratio that connects what you know to what you need
If you know a hypotenuse and need an opposite side, sine uses exactly those sides. If you know the two legs, tangent may avoid calculating the hypotenuse. A complementary angle swaps opposite and adjacent, explaining why sin θ = cos(90° − θ) for acute angles.
Worked example
Relative to angle θ, a right triangle has opposite side 5 and hypotenuse 13. Find sin θ.
- Select the sine ratio: opposite divided by hypotenuse.
- Substitute the two given lengths.
sin θ = 5/13
Watch for this: The adjacent leg is not the hypotenuse, even though the hypotenuse also touches the angle.
Check your understanding
A right triangle has opposite side 3 and adjacent side 4 relative to θ. What is tan θ?
- 3/5
- 4/3
- 4/5
- 3/4
Show answer and explanation
Answer: D. Tangent is opposite/adjacent = 3/4.
Area and volume
Area, surface area, and volume
Select the formula for what is being measured.
Area is two-dimensional
Rectangle: lw; triangle: bh/2; circle: πr². A triangle’s height is perpendicular to its chosen base. Composite areas can be added or subtracted.
Volume is three-dimensional
Prism or cylinder: base area × height. A pyramid or cone has one-third the volume of the corresponding prism or cylinder. Surface area measures the outside faces, not the space inside.
Area counts square units; volume counts cubic units
A rectangle of length l and width w contains lw unit squares. A triangle is half of a rectangle or parallelogram with the same base and perpendicular height, giving bh/2. A prism stacks identical bases through a height, so its volume is base area multiplied by height.
Choose the measured quantity before selecting a formula
Perimeter measures the boundary in length units, surface area measures exterior faces in square units, and volume measures enclosed space in cubic units. A cone or pyramid has one-third the volume of the corresponding cylinder or prism with the same base and height. Identify radius, diameter, and perpendicular height explicitly.
Worked example
Find the volume of a cylinder with radius 3 and height 5.
- The base area is π(3²) = 9π.
- Multiply by height 5.
45π cubic units
Watch for this: A diameter is twice the radius. Convert it before using πr² or a circle-based volume.
Check your understanding
A triangle has base 10 cm and perpendicular height 6 cm. What is its area?
- 30 cm²
- 60 cm²
- 16 cm²
- 300 cm²
Show answer and explanation
Answer: A. Area = (1/2)(10)(6) = 30 square centimeters.
Area and volume
Length, area, and volume scale factors
Distinguish how lengths, areas, and volumes change under similarity.
Square and cube the length factor
If every length is multiplied by k, area is multiplied by k² and volume by k³. This requires similar figures with all corresponding dimensions scaled alike.
Work backward
If the area ratio is 9 for similar figures, the positive length factor is √9 = 3. If the volume ratio is 8, the length factor is ∛8 = 2.
Scaling multiplies every independent dimension
If a similar figure’s lengths are multiplied by k, a rectangle’s area becomes (kl)(kw) = k²lw. A box’s volume becomes (kl)(kw)(kh) = k³lwh. The square or cube arises from multiplying two or three scaled dimensions, not from an arbitrary rule.
Use roots when working backward
A given area ratio must be square-rooted to find a positive length scale factor; a volume ratio must be cube-rooted. These rules require similarity. If only a cylinder’s height doubles while its radius stays fixed, its volume doubles instead of multiplying by 2³.
Worked example
A cube’s edge length is doubled. How does its volume change?
- Every dimension is multiplied by 2.
- Volume scales by 2 × 2 × 2.
The volume is multiplied by 8.
Watch for this: Changing only one dimension does not use the same similarity rule as scaling the entire figure.
Check your understanding
Two similar figures have a length scale factor of 3. What is their area scale factor?
- 3
- 6
- 9
- 27
Show answer and explanation
Answer: C. Area scales by 3² = 9.
Circles
Circle equations and completing the square
Find a circle’s center and radius from its equation.
Standard form
(x − h)² + (y − k)² = r² represents a circle with center (h, k) and radius r > 0. The number on the right is radius squared.
Complete the square
To turn x² + bx into a square, add (b/2)². When changing an equation, add that amount to both sides, or balance it elsewhere on the same side.
A circle consists of points a fixed distance from its center
The distance from (x, y) to center (h, k) is √((x − h)² + (y − k)²). Setting that distance equal to r and squaring yields the circle equation. This explains both the opposite signs inside the parentheses and the squared radius on the right.
Completing the square rebuilds the distance form
The expansion (x + a)² = x² + 2ax + a² shows that half the x-coefficient is the number inside the square. Add its square to complete the expression, balancing the equation by adding the same amount to the other side. Repeat separately for y.
Worked example
Find the center and radius of x² − 6x + y² + 4y = 12.
- Add 9 and 4 to both sides to complete both squares.
- Get (x − 3)² + (y + 2)² = 25.
Center: (3, −2); radius: 5.
Watch for this: Inside the squares, signs are opposite the center coordinates.
Check your understanding
What is the radius of (x + 1)² + (y − 4)² = 36?
- 36
- 6
- 18
- 4
Show answer and explanation
Answer: B. r² = 36, so the positive radius is r = 6.
Circles
Arcs, radians, and circle angles
Connect an angle to its fraction of a circle.
Arc and sector
A central angle θ in degrees gives arc length (θ/360)·2πr and sector area (θ/360)·πr². In radians, arc length is rθ and sector area is r²θ/2. One full turn is 360° = 2π radians.
Circle angle facts
An inscribed angle is half the measure of its intercepted arc. A tangent is perpendicular to the radius at the point of tangency. On the unit circle, coordinates at angle θ are (cos θ, sin θ).
Radians measure an angle using arc length
An angle of 1 radian intercepts an arc whose length equals the radius. In general θ = arc length/radius, so arc length = rθ when θ is in radians. A full circumference is 2πr, which corresponds to 2π radians or 360°.
Central and inscribed angles use different relationships
A central angle has its vertex at the center and matches the degree measure of its intercepted arc. An inscribed angle has its vertex on the circle and measures half its intercepted arc. Sector area uses the same fraction of the full circle as the central angle; arc length uses that fraction of the circumference.
Worked example
A circle of radius 6 has a central angle of 60°. Find the intercepted arc length.
- 60° is 1/6 of a full turn.
- The circumference is 12π.
- Take one-sixth of 12π.
Arc length: 2π
Watch for this: Use rθ only when θ is in radians, and distinguish arc length from arc angle.
Check your understanding
An inscribed angle intercepts an arc measuring 100°. What is the angle?
- 100°
- 200°
- 25°
- 50°
Show answer and explanation
Answer: D. The inscribed angle is 100°/2 = 50°.
Put the ideas into practice
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