All SAT Math Resources
Example Questions
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 #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?
4
32
16
8
2
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.
Example Question #1 : Diameter
The perimeter of a circle is 36 π. What is the diameter of the circle?
3
36
72
18
6
36
The perimeter of a circle = 2 πr = πd
Therefore d = 36
Example Question #282 : Plane Geometry
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?
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?
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?
Write the formula for the circumference.
Substitute the circumference.
Example Question #6 : Diameter
Find the diameter of a circle whose area is .
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,