Circle Geometry

Radius (r): The distance from the center of a circle to any point on the circle. All radii of a circle are equal in length.

Diameter (d): The distance across a circle through its center. The diameter is twice the radius: $d=2r$.

Circumference (C): The distance around a circle. The circumference is related to the diameter by the formula $C=\pi d$ or $C=2\pi r$.

Area (A): The amount of space inside a circle. The area is related to the radius by the formula $A=\pi r^2$.

Note: The formulas for circumference and area are provided on the SAT reference sheet, so you don't need to memorize them.

Chord: A line segment whose endpoints lie on the circle.

Diameter: A chord that passes through the center of the circle. It is the longest possible chord.

Arc: A portion of the circumference of a circle. It is measured in degrees or length.

Sector: A region bounded by two radii and an arc. It is like a "pie slice" of the circle.

Segment: A region bounded by a chord and an arc.

Tangent: A line that touches the circle at exactly one point. The tangent is perpendicular to the radius at the point of tangency.

A central angle is an angle whose vertex is at the center of the circle and whose sides are radii of the circle.

The measure of a central angle is equal to the measure of its intercepted arc.

Example: If a central angle measures 60°, then its intercepted arc also measures 60°.

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle.

The measure of an inscribed angle is half the measure of its intercepted arc.

Example: If an inscribed angle intercepts an arc that measures 80°, then the inscribed angle measures 40°.

Important Relationship: A central angle that intercepts the same arc as an inscribed angle has twice the measure of the inscribed angle.

Example: If an inscribed angle measures 30°, then a central angle intercepting the same arc measures 60°.

Tangent-Chord Angle: An angle formed by a tangent and a chord at the point of tangency measures half the measure of its intercepted arc.

Example: If a tangent-chord angle intercepts an arc that measures 100°, then the angle measures 50°.

Angles Inside a Circle: An angle formed by two chords that intersect inside a circle measures half the sum of the measures of its intercepted arcs.

Example: If two chords intersect inside a circle and intercept arcs that measure 60° and 80°, then the angle formed by the chords measures $\frac{60°+80°}{2}=70°$.

Angles Outside a Circle: An angle formed by two secants, two tangents, or a secant and a tangent that intersect outside a circle measures half the difference of the measures of its intercepted arcs.

Example: If two secants intersect outside a circle and intercept arcs that measure 120° and 40°, then the angle formed by the secants measures $\frac{120°-40°}{2}=40°$.

The length of an arc is proportional to the measure of its central angle. This relationship can be expressed as a proportion:

$\frac{\mathrm{arc\; length}}{\mathrm{circumference}}=\frac{\mathrm{central\; angle}}{360}$

To find the arc length, you can solve this proportion:

$\mathrm{arc\; length}=\frac{\mathrm{central\; angle}}{360}\times \mathrm{circumference}$

Example: If a central angle measures 60° and the circumference of the circle is 12π, then the length of the intercepted arc is $\frac{60}{360}\times 12\pi =2\pi$.

The area of a sector is proportional to the measure of its central angle. This relationship can be expressed as a proportion:

$\frac{\mathrm{sector\; area}}{\mathrm{circle\; area}}=\frac{\mathrm{central\; angle}}{360}$

To find the sector area, you can solve this proportion:

$\mathrm{sector\; area}=\frac{\mathrm{central\; angle}}{360}\times \mathrm{circle\; area}$

Example: If a central angle measures 90° and the area of the circle is 16π, then the area of the sector is $\frac{90}{360}\times 16\pi =4\pi$.

When solving problems involving arc length or sector area, you can set up proportions to find unknown values.

Example: If an arc with a central angle of 60° has a length of 5π, what is the circumference of the circle?

Set up the proportion: $\frac{5}{\pi }=\frac{60}{360}$

Cross-multiply: $5\pi \times 360=60\times C$

Solve for C: $C=\frac{5\pi \times 360}{60}=30\pi$

Therefore, the circumference of the circle is 30π.

Perpendicularity: A tangent is perpendicular to the radius at the point of tangency.

Two Tangents from a Point: If two tangents are drawn from an external point to a circle, they are equal in length.

Example: If point P is outside a circle and tangents PA and PB are drawn to the circle, then PA = PB.

Tangent-Chord Angle: An angle formed by a tangent and a chord at the point of tangency measures half the measure of its intercepted arc.

Example: If a tangent-chord angle intercepts an arc that measures 100°, then the angle measures 50°.

Finding the Length of a Tangent: If a tangent is drawn from an external point to a circle, you can use the Pythagorean theorem to find its length.

Example: If a tangent is drawn from point P to a circle with radius 5, and the distance from P to the center of the circle is 13, then the length of the tangent is $\sqrt{{13}^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

Finding the Distance from a Point to a Circle: If a tangent is drawn from an external point to a circle, you can use the Pythagorean theorem to find the distance from the point to the circle.

Example: If a tangent with length 12 is drawn from point P to a circle with radius 5, then the distance from P to the center of the circle is $\sqrt{{12}^2+5^2}=\sqrt{144+25}=\sqrt{169}=13$.

To convert from degrees to radians, multiply by $\frac{\pi }{180}$.

To convert from radians to degrees, multiply by $\frac{180}{\pi }$.

Example: To convert 60° to radians, multiply by $\frac{\pi }{180}$: $60°\times \frac{\pi }{180}=\frac{\pi }{3}$.

Example: To convert $\frac{\pi }{4}$ radians to degrees, multiply by $\frac{180}{\pi }$: $\frac{\pi }{4}\times \frac{180}{\pi }=45°$.

Common Conversions:

• 30° = $\frac{\pi }{6}$ radians
• 45° = $\frac{\pi }{4}$ radians
• 60° = $\frac{\pi }{3}$ radians
• 90° = $\frac{\pi }{2}$ radians
• 180° = π radians
• 360° = 2π radians

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define the sine, cosine, and tangent functions.

Sine (sin): The y-coordinate of the point where the terminal side of an angle intersects the unit circle.

Cosine (cos): The x-coordinate of the point where the terminal side of an angle intersects the unit circle.

Tangent (tan): The ratio of sine to cosine: $\tan \left(\theta \right)=\frac{\sin \left(\theta \right)}{\cos \left(\theta \right)}$.

Common Values:

• sin(0°) = 0
• cos(0°) = 1
• tan(0°) = 0
• sin(30°) = $\frac{1}{2}$
• cos(30°) = $\frac{\sqrt{3}}{2}$
• tan(30°) = $\frac{1}{\sqrt{3}}$
• sin(45°) = $\frac{1}{\sqrt{2}}$
• cos(45°) = $\frac{1}{\sqrt{2}}$
• tan(45°) = 1
• sin(60°) = $\frac{\sqrt{3}}{2}$
• cos(60°) = $\frac{1}{2}$
• tan(60°) = $\sqrt{3}$
• sin(90°) = 1
• cos(90°) = 0
• tan(90°) is undefined

Note: These values are provided on the SAT reference sheet, so you don't need to memorize them.

The unit circle can be used to solve various trigonometry problems.

Example: If sin(θ) = $\frac{1}{2}$, what is cos(θ)?

From the unit circle, we know that sin(30°) = $\frac{1}{2}$, so θ = 30°.

Therefore, cos(θ) = cos(30°) = $\frac{\sqrt{3}}{2}$.

Example: If cos(θ) = $\frac{1}{2}$, what is tan(θ)?

From the unit circle, we know that cos(60°) = $\frac{1}{2}$, so θ = 60°.

Therefore, tan(θ) = tan(60°) = $\sqrt{3}$.

When solving for missing angles in circles, use the relationships between central angles, inscribed angles, and intercepted arcs.

Example: If an inscribed angle measures 40°, what is the measure of its intercepted arc?

Since the measure of an inscribed angle is half the measure of its intercepted arc, the intercepted arc measures 80°.

Example: If a central angle intercepts an arc that measures 120°, what is the measure of an inscribed angle that intercepts the same arc?

Since the measure of an inscribed angle is half the measure of its intercepted arc, the inscribed angle measures 60°.

When solving for arc lengths or sector areas, set up proportions using the relationships between central angles, arc lengths, and sector areas.

Example: If a central angle measures 45° and the circumference of the circle is 16π, what is the length of the intercepted arc?

Set up the proportion: $\frac{\mathrm{arc\; length}}{16}=\frac{45}{360}$

Cross-multiply: $\mathrm{arc\; length}\times 360=45\times 16\pi$

Solve for arc length: $\mathrm{arc\; length}=\frac{45\times 16\pi }{360}=2\pi$

Therefore, the length of the intercepted arc is 2π.

When solving tangent problems, remember that a tangent is perpendicular to the radius at the point of tangency.

Example: If a tangent is drawn from point P to a circle with radius 6, and the distance from P to the center of the circle is 10, what is the length of the tangent?

Since the tangent is perpendicular to the radius at the point of tangency, we can use the Pythagorean theorem to find the length of the tangent:

$\sqrt{{10}^2-6^2}=\sqrt{100-36}=\sqrt{64}=8$

Therefore, the length of the tangent is 8.