Circles on the Digital SAT: Equations, Arcs, Sectors & Tangents

Circles are an official skill in the SAT Geometry and Trigonometry domain.

College Board says Geometry and Trigonometry contributes about 5–7 questions total across area/volume, lines/angles/triangles, right-triangle trigonometry, and circles. It does not promise a fixed number of circle questions.

Circle equation

Standard form:

[ (x-h)^2+(y-k)^2=r^2 ]

Center:

[ (h,k) ]

Radius:

[ r ]

Example 1

[ (x-4)^2+(y+3)^2=49 ]

Center:

[ (4,-3) ]

Radius:

[ 7 ]

The sign inside the parentheses is opposite the coordinate.

Find an equation from center and radius

Center ((-2,5)), radius 6:

[ (x+2)^2+(y-5)^2=36 ]

Complete the square

The SAT may give a circle in expanded form.

Example 2

[ x^2+y^2-6x+8y-11=0 ]

Group:

[ (x^2-6x)+(y^2+8y)=11 ]

Complete each square:

[ x^2-6x+9=(x-3)^2 ]

[ y^2+8y+16=(y+4)^2 ]

Completing the square here uses the same technique tested in quadratic equations problems.

Add 9 and 16 to both sides:

[ (x-3)^2+(y+4)^2=36 ]

Center:

[ (3,-4) ]

Radius:

[ 6 ]

Diameter and radius

[ d=2r ]

A surprisingly common error is reading (r^2=36) and saying the radius is 36.

Radius = 6.

Circumference

[ C=2\pi r ]

or

[ C=\pi d ]

Area

[ A=\pi r^2 ]

Arc length

If the central angle is (\theta) degrees:

[ \text{arc length}=\frac{\theta}{360}(2\pi r) ]

Example 3

Radius 9, central angle 80°:

[ \frac{80}{360}(18\pi) ]

[ =4\pi ]

Sector area

[ \text{sector area}=\frac{\theta}{360}(\pi r^2) ]

With radians:

[ \text{sector area}=\frac12r^2\theta ]

when (\theta) is in radians.

Radians

Key conversion:

[ 180^\circ=\pi\text{ radians} ]

Examples:

[ 90^\circ=\frac{\pi}{2} ]

[ 60^\circ=\frac{\pi}{3} ]

[ 45^\circ=\frac{\pi}{4} ]

Arc length in radians

If (\theta) is in radians:

[ s=r\theta ]

Example 4

Radius 5, central angle (2\pi/3):

[ s=5\left(\frac{2\pi}{3}\right)=\frac{10\pi}{3} ]

Tangent-radius theorem

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

That creates a 90° angle.

This often turns a circle problem into a right-triangle problem. Finding where a tangent line meets a circle is really a systems of equations problem in disguise.

Angles in circles

Central angle

Vertex at center.

Its degree measure equals the measure of its intercepted arc.

Inscribed angle

Vertex on circle.

Its measure is half the measure of its intercepted arc.

If an inscribed angle intercepts a 120° arc:

[ angle=60^\circ ]

Diameter and right angles

An angle inscribed in a semicircle is a right angle.

If a triangle is formed with one side as a diameter and the third point on the circle, the angle opposite the diameter is 90°.

Distance from center to a point

If a point lies on a circle, its distance from the center equals the radius.

You can use the distance formula:

[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]

Desmos and circles

Desmos can graph:

[ (x-2)^2+(y+1)^2=25 ]

This can help with:

• visualizing center/radius;
• line-circle intersections;
• checking tangent/intersection behavior.

But if the question asks only for radius from standard form, graphing is slower than reading it. Line-circle intersections lean on the same slope-intercept skills covered in linear equations.

Common SAT traps

Trap 1: wrong center signs

[ (x+3)^2+(y-4)^2=16 ] center = ((-3,4)).

Trap 2: radius squared

Right side 81 means radius 9.

Trap 3: arc vs sector

Arc = length. Sector = area.

Trap 4: degrees formula used with radians

Know which angle unit you have.

Trap 5: diameter used as radius

Check labels.

Original mini-practice

1

What is the center of [ (x+5)^2+(y-2)^2=64 ]?

2

A circle has radius 12. What is the length of an arc subtended by a 75° central angle?

3

A sector has radius 6 and central angle 120°. What is its area?

4

A circle is represented by: [ x^2+y^2+4x-10y+13=0 ] What is its radius?

5

An inscribed angle intercepts an arc measuring 146°. What is the angle measure?

Answers

1. (-5, 2)

2. (5\pi)
   (\frac{75}{360}(24\pi)=5\pi).

3. (12\pi)
   (\frac{120}{360}(36\pi)=12\pi).

4. 4
   Complete square: ((x+2)^2+(y-5)^2=16).

5. 73°
   Inscribed angle = half intercepted arc.

What to learn next

• right triangles and trigonometry;
• coordinate geometry;
• distance formula;
• radians/unit circle.

See the rest of our SAT Math guides.

Official references

• College Board, Geometry and Trigonometry: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/geometry-trigonometry

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