Radians and the arc and sector formulas in radians, the unit circle and the signs of sine and cosine by quadrant, the identity sin² + cos² = 1, inscribed angles and tangents, completing the square to find a circle's center and radius, congruence and similarity conditions, and how scaling a figure scales its area and volume.
The last geometry questions on a hard module take the facts from the earlier lessons and ask them in a form that requires one more idea: an angle in radians instead of degrees, a trigonometric value for an angle that is not in a triangle, a circle whose equation has not been arranged for you, an inscribed angle instead of a central one, or a solid whose dimensions have all been scaled.
Each of those has a single fact behind it. A full turn is 2 π radians, so every degree formula becomes a radian formula by replacing 360 with 2 π. Sine and cosine are the coordinates of a point on the unit circle, which fixes their signs. The Pythagorean theorem on that circle is sin² + cos² = 1. An inscribed angle is half the central angle on the same arc. Completing the square turns any circle equation into center-radius form. And scaling every length by k scales areas by k² and volumes by k³.
These are the six ideas of this lesson, each with the question it unlocks.
Convert with π = 180 degrees, and use s = r theta only in radians
To convert degrees to radians multiply by π/180; to convert back multiply by 180/π. Keep the answer as a multiple of π: 150 degrees is 5 π/6. In radians the arc length is simply r times theta and the sector area is (1/2) r² theta, which is faster than the degree fractions, but only if theta is in radians. An angle like 3 π/4 in a formula with 360 in it is the classic mismatch.
Place the angle on the unit circle to get the sign
Sine is the y-coordinate and cosine the x-coordinate of the point where the angle's ray meets the unit circle. So both are positive in the first quadrant; only sine is positive in the second; both are negative in the third; only cosine is positive in the fourth. When a question gives sin theta and says which quadrant theta is in, that quadrant decides the sign of cos theta before any arithmetic.
Use sin² + cos² = 1 to move between the two
Given one of sine or cosine, the other is a square root away: cos theta = ±√(1 - sin² theta), with the sign from the quadrant. This is the Pythagorean theorem for the unit circle, and it also gives tan theta = sin theta / cos theta. A question phrased entirely in terms of a letter, such as "if sin theta = a, what is cos theta", wants √(1 - a²) for an acute angle.
Halve a central angle to get an inscribed one, and look for the right angle a tangent or a diameter makes
An inscribed angle is half the central angle that opens onto the same arc, so an angle inscribed in a semicircle is a right angle. A radius drawn to the point where a tangent touches the circle is perpendicular to the tangent. Both facts plant a right triangle in the figure, and then the Pythagorean theorem and the trigonometric ratios take over.
Complete the square in x and in y separately
x² + y² + Dx + Ey + F = 0 becomes (x + D/2)² + (y + E/2)² = D²/4 + E²/4 - F. Group the x-terms, add the square of half the x-coefficient; do the same for y; add both to the right side along with -F. The center is (-D/2, -E/2) and the radius is the square root of the right side. Then apply the k² and k³ rules if the question scales anything.
A radian is the angle whose arc equals the radius, and a full circle is 2 π radians, so π radians equals 180 degrees. Conversions follow: 30 degrees is π/6, 45 is π/4, 60 is π/3, 90 is π/2, 150 is 5 π/6, 210 is 7 π/6, and in general degrees times π/180 gives radians. The advantage of radians is that the circle formulas lose their fractions: an arc cut by a central angle theta in radians has length s = r theta and the sector has area (1/2) r² theta. A radius of 6 and an angle of 5 π/6 give an arc of 5 π and a sector of 15 π. The SAT reference sheet states that a circle has 2 π radians, and questions ask you to convert, to find an arc length from a radian measure, or to find a radian measure from an arc length and radius.
An angle in standard position has its vertex at the origin and one ray along the positive x-axis; the other ray meets the unit circle at a point whose coordinates are (cos theta, sin theta). That definition extends sine and cosine beyond acute angles and fixes their signs by quadrant: in the second quadrant x is negative and y positive, so cosine is negative and sine positive; in the third both are negative; in the fourth cosine is positive and sine negative. It also explains the special values: at 60 degrees, or π/3, the point is (1/2, √3/2), so cos(π/3) = 1/2 and sin(π/3) = √3/2, the 30-60-90 triangle placed on the circle. The cofunction identity sin x = cos(π/2 - x) is the same statement in radians, so sin(π/6) = cos(π/3) = 1/2.
Because the point (cos theta, sin theta) lies on the unit circle, its coordinates satisfy x² + y² = 1, which is the identity sin² theta + cos² theta = 1. Given one of the two, the other follows up to sign: if sin theta = 3/5 and theta is in the second quadrant, then cos² theta = 1 - 9/25 = 16/25 and cos theta = -4/5, negative because the second quadrant has negative x. Tangent is sine over cosine, here -3/4. A question written with a letter, "if sin theta = a for an acute angle theta, express cos theta", has answer √(1 - a²), and the same identity underlies questions that ask for the value of an expression like 1 - sin² theta, which is cos² theta.
Circles carry a short list of angle and tangent facts. A central angle equals the measure of its arc. An inscribed angle, with its vertex on the circle, is half the central angle on the same arc, so all inscribed angles on the same arc are equal and an angle inscribed in a semicircle is 90 degrees. A tangent line touches the circle at one point and is perpendicular to the radius drawn to that point, so a tangent segment from an external point, the radius to the point of tangency, and the segment from the external point to the center form a right triangle: with radius 5 and the external point 13 from the center, the tangent segment is 12. Two tangent segments from the same external point are equal. A chord's perpendicular bisector passes through the center.
The equation of a circle is often given expanded, x² + y² + Dx + Ey + F = 0, and reading its center and radius requires completing the square in both variables. For x² + y² - 4x + 6y - 3 = 0, group as (x² - 4x) + (y² + 6y) = 3, add 4 and 9 to both sides, and get (x - 2)² + (y + 3)² = 16: center (2, -3), radius 4. If the right side comes out negative the equation describes no points at all, and if it is zero, a single point. Questions also run the process backwards, asking which expanded equation represents a circle with a given center and radius, which is a matter of expanding the standard form.
Similar figures have all lengths in a common ratio k, and that ratio governs the other measurements: areas are multiplied by k² and volumes by k³. Doubling the radius of a sphere multiplies its volume by 8; increasing a circle's radius by 20 percent multiplies its area by 1.2² = 1.44, a 44 percent increase; halving every dimension of a box divides its volume by 8. Conversely, if two similar solids have volumes in the ratio 27 to 8 their lengths are in the ratio 3 to 2. Triangles are congruent, identical in every measurement, under SSS, SAS, ASA and AAS, but not under SSA or AAA; AAA gives similarity, which needs only two matching angles. The triangle inequality, that any side is shorter than the sum of the other two, decides whether three lengths can form a triangle at all.
Composite figures at this level are handled by addition and subtraction: the area of a shaded region is the area of the outer figure minus the inner, the volume of a pipe is the outer cylinder minus the inner one, and a shape cut from a rectangle leaves the rectangle's area minus the cutout. The order of operations is the pitfall: compute each simple piece completely, with its own radius or dimensions, before combining, and keep π symbolic until the end so the answer choices can be matched.
The unit circle: sine and cosine are coordinates
The angle π/3 (60 degrees) in standard position meets the unit circle at (1/2, √3/2), so cos(π/3) = 1/2 and sin(π/3) = √3/2. Since the point is on the circle, (1/2)² + (√3/2)² = 1: that is sin² + cos² = 1. In the second quadrant x is negative, so cosine is negative there.
An inscribed angle is half the central angle
Points A and B are on the circle with center O, and C is a third point on the circle. Central angle AOB opens onto arc AB and measures 100 degrees; inscribed angle ACB opens onto the same arc and measures 50 degrees. Move C anywhere on the major arc and the inscribed angle stays 50.
A tangent is perpendicular to the radius
Line PT is tangent to the circle at T, and O is the center. Because OT is perpendicular to PT, triangle OTP is a right triangle with the right angle at T. With OT = 5 and OP = 13, the tangent segment is PT = √(169 - 25) = 12.
Degrees, radians and the circle formulas
Replace 360 with 2 π and the degree fractions become the radian formulas. The radian formulas are only valid with theta in radians.
| Degrees | Radians | Arc length, r = 6 | Sector area, r = 6 |
|---|---|---|---|
| 30° | π/6 | 6 x π/6 = π | (1/2)(36)(π/6) = 3 π |
| 90° | π/2 | 3 π | 9 π |
| 150° | 5 π/6 | 5 π | 15 π |
| 210° | 7 π/6 | 7 π | 21 π |
| theta degrees | theta x π/180 | r theta (radians) = (theta/360) x 2 π r | (1/2) r² theta (radians) = (theta/360) x π r² |
Scale a figure by k: lengths x k, areas x k², volumes x k³
Every length, including radius, height, perimeter and circumference, scales by k. Areas, including surface area, scale by k². Volumes scale by k³. A percent change in a length is a factor of (1 + r/100).
| Change to every length | k | Perimeter or circumference | Area | Volume |
|---|---|---|---|---|
| Doubled | 2 | x2 | x4 | x8 |
| Tripled | 3 | x3 | x9 | x27 |
| Halved | 1/2 | x1/2 | x1/4 | x1/8 |
| Increased 20% | 1.2 | +20% | x1.44: +44% | x1.728: +72.8% |
| Volumes in ratio 27 : 8 | 3/2 | 3 : 2 | 9 : 4 | 27 : 8 |
Example 1: Convert to radians and find an arc length
A circle has radius 6. What is the length of the arc cut off by a central angle of 150 degrees? Give the answer in terms of π.
The degree route gives the same thing: (150/360)(12 π) = 5 π. Use whichever the question's units suggest, but never mix them.
Answer: 5 π
Example 2: Find cosine from sine using the quadrant
If sin theta = 3/5 and theta is between π/2 and π, what is the value of cos theta?
Without the quadrant information the answer would be ±4/5; the interval given is what selects the negative root.
Answer: cos theta = -4/5
Example 3: Complete the square to find a circle's center and radius
The equation x² + y² - 4x + 6y - 3 = 0 represents a circle in the xy-plane. What are its center and radius?
The right side must come out positive for the equation to be a circle; 16 does.
Answer: Center (2, -3), radius 4
Example 4: Inscribed angle from a central angle
Points A, B and C lie on a circle with center O. The central angle AOB measures 100 degrees and C lies on the major arc AB. What is the measure of angle ACB?
If C were on the minor arc instead, angle ACB would open onto the major arc of 260 degrees and measure 130 degrees. Where C sits matters.
Answer: 50 degrees
Example 5: Scaling and volume
The radius of a sphere is increased by 20 percent. By what percent does the volume increase?
Not 60 percent (3 x 20) and not 20 percent. Check with the formula: (4/3) π (1.2r)³ = 1.728 x (4/3) π r³.
Answer: 72.8 percent
Check your understanding 1
What is the radian measure of an angle of 210 degrees?
Check your understanding 2
A circle has radius 4. What is the length of an arc cut off by a central angle of 3 π/4 radians?
Check your understanding 3
If sin theta = 3/5 and π/2 < theta < π, what is the value of cos theta?
Check your understanding 4
The equation x² + y² - 4x + 6y - 3 = 0 represents a circle in the xy-plane. What is the radius of the circle?
Check your understanding 5
Points A, B and C lie on a circle with center O, with C on the major arc AB. If the central angle AOB measures 100 degrees, what is the measure of the inscribed angle ACB?
Check your understanding 6
The radius of a sphere is doubled. The volume of the new sphere is how many times the volume of the original?
Check your understanding 7
Line segment PT is tangent to a circle with center O at point T. If the radius of the circle is 5 and OP = 13, what is the length of PT?
Check your understanding 8
For an acute angle theta, sin theta = a. Which expression is equal to cos theta?
Check your understanding 9
Which of the following is equal to sin(π/6)?
Using a radian measure in a formula built for degrees, or the reverse. The 360 in a formula means degrees; r theta means radians.
Taking the positive square root for cosine or sine without checking the quadrant.
Confusing cos² theta with cos theta: 1 - sin² theta is the square of the cosine.
Treating an inscribed angle as equal to its central angle, or halving when the point is on the minor arc so the angle is actually half of the major arc.
Completing the square by adding to one side only, or reading the right side as the radius instead of its square.
Scaling area by k and volume by k, or by 2k and 3k: it is k² and k³.
Assuming a tangent segment is the hypotenuse; the segment from the external point to the center is.
Same topic, a different level:
Practise this: drills for this topic in the question bank.