SAT Math : Geometry

Study concepts, example questions & explanations for SAT Math

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Example Questions

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 #281 : Plane Geometry

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:

4

32

16

8

2

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.

Example Question #1 : Diameter

The perimeter of a circle is 36 π.  What is the diameter of the circle?

Possible Answers:

3

36

72

18

6

Correct answer:

36

Explanation:

The perimeter of a circle = 2 πr = πd

Therefore d = 36

Example Question #282 : Plane Geometry

Sat_math_picture

If the area of the circle touching the square in the picture above is , what is the closest value to the area of the square?

Possible Answers:

Correct answer:

Explanation:

Obtain the radius of the circle from the area.

Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be , , and . The radii of the circle (from the center to the corners of the square) will be 9. The hypotenuse (side of the square) must be .

The area of the square is then .

Example Question #7 : Diameter

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle? 

Possible Answers:

Correct answer:

Explanation:

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr2.

Example Question #5 : Diameter

The circumference of the circle is .  What is the diameter?

Possible Answers:

Correct answer:

Explanation:

Write the formula for the circumference.

Substitute the circumference.

Example Question #6 : Diameter

Find the diameter of a circle whose area is .

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for the area of a circle to find the radius, and then multiply it by 2 to find the diameter. Thus,

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