Everything about the Circle on the SAT
What is a circle?
A circle is the collection of all points equidistant (at equal distance) from a given point.
- This given point is called the center. A circle is named after its center point.
- The distance from the center to any point on the circle is called the radius, (r). It is the most important measurement in a circle. We can figure out all the other characteristics if we know the radius.
- The diameter(d) of a circle is a line segment whose endpoints lie on the circle and which passes through the center. It is twice as long as the radius (d = 2r)
- A chord is a line segment whose endpoints lie on the circle, but it does not necessarily pass through the center.
Therefore, a diameter is always a chord, but a chord is not always a diameter.
- A line that touches the circle at only one point is called a tangent
- An extended line that intersects the circle at two points is called a secant
- Any connected part of a circle is called an arc
- A region bounded by two radii and an arc lying between the radii is called a sector
- a region, not containing the center, bounded by a chord and an arc lying between the chord’s endpoints is called a segment
Properties of Tangents
- The radius whose endpoint is the intersection point of the tangent line and the circle is always perpendicular to the tangent line.
- Every point in space outside the circle can extend exactly two tangent lines to the circle. The distance from the origin of the two tangents to the points of tangency are always equal. In the figure below, XY = XZ.
An example problem: What is the area of triangle QRS, if RS is tangent to the circle Q?
Now, we know that if RS is a tangent to Q, then QR is perpendicular to RS, and therefore QRS is a 30-60-90 triangle. Given that QR = 4, we know that RS = 4 tan 60º , and the area of triangle QRS
- When two chords intersect each other inside a circle, the products of their segments are equal. Let us illustrate this with a diagram.
Each chord is cut into two segments at the point of intersection. Euclid’s Intersecting Chord theorem states that a x b = c x d always.
Central Angles and Inscribed Angles
- An angle whose vertex is the center of the circle is called a central angle. The degree of the circle (the slice of pie) cut by a central angle is equal to the measure of the angle. If a central angle is 30º, then it cuts a 30º arc in the circle.
- An inscribed angle is an angle formed by two chords in a circle that originate from a single point.
- If an inscribed angle and a central angle cut out the same arc in a circle, the central angle will be twice as large as the inscribed angle.
In the figure below, ∠AOC is a central angle, whereas ∠ABC is an inscribed angle. If ∠AOC = 60º, then ∠ABC = 30º.
Circumference of a Circle
The circumference is the perimeter of the circle. The formula for circumference is:
C = 2πr
where r is the radius. In other words, circumference of a circle is the length of the 360º arc that forms the circle.
Since diameter of a circle is equal to twice the radius, the formula can also be written C = πd, where d is the diameter.
360º = 2π radians
Radian is the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. Therefore, 1 radian = 360/π degrees
By picking any two points on a circle, two arcs are created: a major arc, which is by definition the longer arc, and a minor arc, which is the shorter one.
- An arc named with only its two endpoints, such as arc BC, will always refer to a minor arc. A minor arc has a measure that is less than 180°
- A 180 º arc is called a semi-circle
- An arc may also be named with three points: the two endpoints and a third point that the arc passes through. So, arc CDB has endpoints at C and B and passes through point D. Three points are generally used to name a major arc. A major arc has a measure of 180° or more.
Since the degree of an arc is defined by the central or inscribed angle that intercepts the arc’s endpoints, you need only know the measure of either of those angles and the measure of the radius of the circle to calculate the arc length. The arc length formula is:
where n is the measure of the degree of the arc, r is the radius and C is the circumference of the circle.
Let us consider an example using this concept:
Circle D has radius 6. What is the length of minor arc AB?
In order to figure out the length of arc AB, we need to know the radius of the circle and the measure of ∠ADC, which is the central angle that intercepts the endpoints of AB. Using the information provided in the figure and the fact that the three angles of a triangle must add up to 180 º,
Area of a Circle
The area of a circle depends on the radius of the circle. The formula for area is:
A = πr²
where r is the radius. If we know the radius, we can always find the area.
Area of a Sector
The area of a sector is related to the area of a circle the same way that the length of an arc is related to circumference. To find the area of a sector, simply find what fraction of 360º the sector comprises and multiply this fraction by the area of the circle.
Area of Sector = n/360 x πr²
where n is the measure of the central angle which forms the boundary of the sector, and r is the radius.
Equation of a circle
In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that
(x – a)² + (y – b)² = r²
This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|.
If the circle is centered at the origin (0, 0), then the equation simplifies to
x² + y² = r²
The equation can be written in parametric form using the trigonometric functions sine and cosine as
x = a + rcosθ
y = b + rsinθ
where θ is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x-axis.