All SAT Math Resources
Example Questions
Example Question #21 : Circles
What is the angle of a sector that has an arc length of on a circle of diameter ?
The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:
For your data, this means,
Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:
Rounded to the nearest hundredth, this is .
Example Question #6 : How To Find The Angle Of A Sector
Figure NOT drawn to scale.
Refer to the above diagram. is a semicircle. Evaluate .
Insufficient information is given to answer the question.
An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, , which intercepts the semicircle , is such an angle. Consequently, , and is a right triangle. The acute angles of are complementary, so
The measure of inscribed is
.
An inscribed angle of a circle intercepts an arc of twice its degree measure, so
.
Example Question #271 : Sat Mathematics
Figure NOT drawn to scale
Refer to the above figure. is a diameter of the circle. Evaluate .
is a diameter, so is a semicircle, and
,
or, equivalently,
In terms of , since ,
and , being a secant segment and a tangent segment to a circle, respectively, intercept two arcs such that the measure of the angle that the segments form is equal to one-half the difference of the measures of the intercepted arcs - that is,
Setting , , and :
Example Question #31 : Circles
Refer to the above diagram. Evaluate the measure of .
The total measure of the arcs that comprise a circle is , so from the above diagram,
Substituting the appropriate expression for each arc measure:
Therefore,
and
The measure of the angle formed by the tangent segments and , which is , is half the difference of the measures of the arcs they intercept, so
Substituting:
Example Question #11 : How To Find The Angle Of A Sector
Figure NOT drawn to scale.
The above figure shows a quadrilateral inscribed in a circle. Evaluate .
The question cannot be answered from the information given.
If a quadrilateral is inscribed in a circle, then each pair of its opposite angles are supplementary - that is, their degree measures total .
and are two such angles, so
Setting and , and solving for :
,
the correct response.
Example Question #281 : Plane Geometry
Figure NOT drawn to scale.
The above figure shows a quadrilateral inscribed in a circle. Evaluate .
The question cannot be answered from the information given.
The question cannot be answered from the information given.
If a quadrilateral is inscribed in a circle, then each pair of its opposite angles are supplementary - that is, their degree measures total .
and are two such angles, so
Setting and , and solving for :
,
The statement turns out to be true regardless of the value of . Therefore, without further information, the value of cannot be determined.
Example Question #281 : Sat Mathematics
Figure NOT drawn to scale.
The above figure shows a quadrilateral inscribed in a circle. Evaluate .
If a quadrilateral is inscribed in a circle, then each pair of its opposite angles are supplementary - that is, their degree measures total .
and are two such angles, so
Setting and , and solving for :
,
the correct response.
Example Question #11 : How To Find The Angle Of A Sector
Figure NOT drawn to scale.
Refer to the above diagram. is a diameter. Evaluate
is a diameter, so is a semicircle - therefore, . By the Arc Addition Principle,
If we let , then
,
and
If a secant and a tangent are drawn from a point to a circle, the measure of the angle they form is half the difference of the measures of the intercepted arcs. Since and are such segments intercepting and , it holds that
Setting , , and :
The inscribed angle that intercepts this arc, , has half this measure:
.
This is the correct response.
Example Question #12 : How To Find The Angle Of A Sector
Figure NOT drawn to scale.
In the above figure, is a diameter. Also, the ratio of the length of to that of is 7 to 5. Give the measure of .
The measure of cannot be determine from the information given.
is a diameter, so is a semicircle, which has measure . By the Arc Addition Principle,
If we let , then, substituting:
,
and
the ratio of the length of to that of is 7 to 5; this is also the ratio of their degree measures; that is,
Setting and :
Cross-multiply, then solve for :
, and
If a secant and a tangent are drawn from a point to a circle, the measure of the angle they form is half the difference of the measures of the intercepted arcs. Since and are such segments whose angle intercepts and , it holds that:
Example Question #1 : Diameter And Chords
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
32
4
16
2
8
16
Set the area of the circle equal to four times the circumference πr2 = 4(2πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.