On the test
Geometry and Trigonometry is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I think about circles, I picture a bicycle wheel. Every spoke from the centre to the rim has the same length. That one fact, the same distance from the middle to every point on the edge, is what makes a circle a circle. If you know the spoke length, you know how far the wheel rolls in one turn and how much rubber the tyre covers. Cut a pizza into equal slices and each slice is just a fraction of the whole wheel. You do not start again for each slice. You take the same fraction of the whole. The SAT circle questions work the same way. They give you a radius, an angle or an equation, and you find the length, the area or the centre by using the same fixed ideas.
Circles can look scary because of the symbols, but they are made of just a few simple ideas: the radius, a fraction of the whole, and one equation. You can learn each one step by step.
What is inside this part
Using C = 2πr and A = πr² to move between radius, length around and space inside.
Finding radius from area · Circumference from area · Diameter and radius
Finding a fraction of the circumference or area using the central angle.
Arc length · Sector area · Radian measure
Reading or writing the equation (x − h)² + (y − k)² = r² from a centre, radius or points.
Centre and radius from the equation · Equation from a diameter · Completing the square
Using circle angle rules and the fact that a tangent is perpendicular to the radius.
Inscribed and central angles · Tangent with a right triangle
How to know a question is from this part
A circle is described by a radius, an angle, a slice, or an equation with x² and y².
- Words like radius, diameter, arc, sector, central angle, tangent, chord
- An equation with both x² and y² and the same coefficient
- 'In the xy-plane, the circle has centre ... and radius ...'
- Answers that contain π
- Angles written in radians, like π/3
The lesson
1. The parts of a circle
The centre is the middle point. The radius (r) is the distance from the centre to any point on the circle. The diameter is a line across the circle through the centre, and it is twice the radius: d = 2r.
The circumference is the distance around the circle. The area is the space inside it. A chord joins two points on the circle. An arc is a piece of the circle's edge. A sector is a slice, like a piece of pizza, made by two radii and an arc. A tangent is a line that touches the circle at exactly one point.
2. Circumference and area
Circumference: C = 2πr (or πd). Area: A = πr². π is about 3.14, but on the SAT, you often leave it as π in the answer.
Do not mix them: circumference is a length (units like cm), area is a surface (units like cm²). If the area is 49π, then r² = 49, so r = 7 and the circumference is 14π.
A circle has an area of 49π. What is its circumference?
Show the steps
- πr² = 49π, so r² = 49 and r = 7.
- C = 2πr = 2 × π × 7 = 14π.
14π
3. Arc length and sector area
A slice is a fraction of the whole circle. That fraction is the central angle divided by 360° (a full turn). Arc length = (angle/360) × 2πr. Sector area = (angle/360) × πr².
So first find the fraction, then take that fraction of the whole circumference or whole area. A 90° angle is 1/4 of the circle, a 60° angle is 1/6, and 180° is 1/2.
A sector has a central angle of 60° and radius 6. What is its area?
Show the steps
- Fraction = 60/360 = 1/6.
- Whole area = π × 6² = 36π.
- Sector area = 36π ÷ 6 = 6π.
6π
4. Radians
Sometimes angles are in radians. A full turn is 2π radians, which is 360°. So π radians = 180°. To change radians to degrees multiply by 180/π. To change degrees to radians multiply by π/180.
With radians the arc length is easy: s = rθ, where θ is in radians. Example: π/3 radians is 60°, since (π/3)(180/π) = 60.
5. The equation of a circle
A circle with centre (h, k) and radius r has the equation (x − h)² + (y − k)² = r². Notice the minus signs: (x − 3)² means h = 3, and (y + 2)² means k = −2 (because y + 2 is y − (−2)).
The number on the right is r², not r. If it says = 25, the radius is 5.
A circle has equation (x − 3)² + (y + 2)² = 25. What are the centre and radius?
Show the steps
- x − 3 gives h = 3. y + 2 = y − (−2) gives k = −2.
- Centre is (3, −2).
- r² = 25, so r = 5.
Centre (3, −2), radius 5
6. Completing the square
Sometimes the equation is spread out, like x² + y² − 6x + 8y = 0. To read the centre and radius you must rewrite it in the standard form. Group x terms and y terms, then complete the square: take half of the x-coefficient, square it, and add it to both sides. Do the same for y.
For −6x: half of −6 is −3, squared is 9. For 8y: half of 8 is 4, squared is 16. Add 9 and 16 to both sides: (x − 3)² + (y + 4)² = 25. So the centre is (3, −4) and the radius is 5.
A circle has the equation x² + y² + 4x − 10y + k = 0 and radius 6. Find k.
Show the steps
- Complete the square: (x + 2)² + (y − 5)² = 4 + 25 − k = 29 − k.
- The radius is 6 so 29 − k = 36.
- k = −7.
k = −7
7. Equations from points
If you are given two ends of a diameter, the centre is the midpoint: average the x values and average the y values. The radius is half of the distance between the two points. The distance formula is √((x₂ − x₁)² + (y₂ − y₁)²).
Careful: the equation needs r², so you may not need a square root at the end.
A circle has a diameter with ends at (−2, 3) and (6, 7). Write its equation.
Show the steps
- Centre = ((−2 + 6)/2, (3 + 7)/2) = (2, 5).
- Diameter = √(8² + 4²) = √80, so r² = (√80 ÷ 2)² = 80/4 = 20.
- Equation: (x − 2)² + (y − 5)² = 20.
(x − 2)² + (y − 5)² = 20
8. Tangents and inscribed angles
A radius drawn to the point where a tangent touches the circle makes a 90° angle with the tangent. That gives a right triangle, so you can use Pythagoras.
An inscribed angle (its corner is on the circle) is half of the central angle that opens to the same arc. A central angle equals its arc in degrees.
9. Common traps
Trap 1: reading the right side as r instead of r². Trap 2: getting the sign of the centre wrong. Trap 3: using the diameter as the radius. Trap 4: taking the fraction of the circumference when the question asks for the area, or the other way round.
Trap 5: forgetting to move the numbers back to the right side when completing the square.
See it
Read h and k with the opposite sign from the brackets. The right side is r squared.
| What | Formula |
|---|---|
| Diameter | 2r |
| Circumference | 2πr |
| Area | πr² |
| Arc length | (angle ÷ 360) × 2πr |
| Sector area | (angle ÷ 360) × πr² |
| Equation | (x − h)² + (y − k)² = r² |
| Radians to degrees | × 180/π |
- 1Find the central angle
- 2Divide by 360 to get the fraction
- 3Find the whole circumference or whole area
- 4Multiply by the fraction
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