SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

varsity tutors app store varsity tutors android store varsity tutors ibooks store

Example Questions

Example Question #21 : Circles

What is the angle of a sector that has an arc length of   on a circle of diameter  ?

Possible Answers:

Correct answer:

Explanation:

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

Inscribed angle

Figure NOT drawn to scale.

Refer to the above diagram. is a semicircle. Evaluate .

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

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

Secant 2Figure NOT drawn to scale

Refer to the above figure.  is a diameter of the circle. Evaluate .

Possible Answers:

Correct answer:

Explanation:

 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

Secant

Refer to the above diagram. Evaluate the measure of .

Possible Answers:

Correct answer:

Explanation:

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

Inscribed quad

Figure NOT drawn to scale.

The above figure shows a quadrilateral inscribed in a circle. Evaluate .

Possible Answers:

The question cannot be answered from the information given. 

Correct answer:

Explanation:

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

Inscribed quad

Figure NOT drawn to scale.

The above figure shows a quadrilateral inscribed in a circle. Evaluate .

Possible Answers:

The question cannot be answered from the information given. 

Correct answer:

The question cannot be answered from the information given. 

Explanation:

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

Inscribed quad

Figure NOT drawn to scale.

The above figure shows a quadrilateral inscribed in a circle. Evaluate .

Possible Answers:

Correct answer:

Explanation:

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

Secant 2

Figure NOT drawn to scale.

Refer to the above diagram.  is a diameter. Evaluate 

Possible Answers:

Correct answer:

Explanation:

  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

Secant 3Figure 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 

Possible Answers:

The measure of  cannot be determine from the information given.

Correct answer:

Explanation:

 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?

Possible Answers:

32

4

16

2

8

Correct answer:

16

Explanation:

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 = 16.

Learning Tools by Varsity Tutors