Understand circle equations, arc length, sector area, central and inscribed angles, tangent lines, and circle theorems tested on the SAT.
How can I study Circles effectively?โพ
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Circles study guide free?โพ
Yes โ all study notes, flashcards, and practice problems for Circles on Study Mondo are 100% free. No account is needed to access the content.
What course covers Circles?โพ
Circles is part of the SAT Prep course on Study Mondo, specifically in the Additional Topics in Math section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Circles?
C=2ฯr=ฯd
Diameter
d=2r
Arc length
L=360ฮธโร2ฯr
Sector area
A=360ฮธโรฯr2
Equation of a Circle
Standard Form
(xโh)2+(yโk)2=r2
Center: (h,k)
Radius: r
General Form
x2+y2+Dx+Ey+F=0
To convert to standard form: complete the square for both x and y.
Common mistake: Using the diameter (14) instead of the radius (7) in the formula.
2Problem 2medium
โ Question:
What is the center and radius of the circle (x+3)2+(yโ5)2=36?
๐ก Show Solution
Standard form:(xโh)2+(yโk)
3Problem 3medium
โ Question:
A sector of a circle with radius 10 has a central angle of 72ยฐ. What is the area of the sector?
๐ก Show Solution
Sector area formula:A=360ฮธโรฯr2
A=36072โรฯ(10)
Answer:20ฯ square units (approximately 62.83)
Check: 72ยฐ is 51โ of 360ยฐ, so the sector is 51โ of the full circle area. โ
4Problem 4hard
โ Question:
Convert to standard form and find the center and radius:
x2+y2+8xโ10y+16=0
๐ก Show Solution
Step 1: Group x terms and y terms, move constant:
(x2+
5Problem 5expert
โ Question:
A circle with center (2,3) is tangent to the line y=โ1. What is the equation of the circle?
๐ก Show Solution
Step 1: "Tangent to the line y=โ1" means the circle touches this horizontal line at exactly one point. The distance from the center to the line equals the radius.
Step 2: The distance from center (2,3) to the line y=โ1 is:
r=โฃ3โ
Step 3: Write the equation:
(xโ2)2+(yโ3)2=16
Check: The closest point on the circle to y=โ1 is directly below the center: (2,3โ4)=(2,โ1). This point is on the line โ
Answer:(xโ2)2+(yโ3)2=16
โพ
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.
4
y
+
4)=
12+
9+
4
2
=
r2
Compare with (x+3)2+(yโ5)2=36:
(x+3)2=(xโ(โ3))2, so h=โ3(yโ5)2, so k=5r2=36, so r=6
Center:(โ3,5)Radius:6
Key: Watch the signs! (x+3)2 means the center x-coordinate is โ3, not +3.
2
=
51โร
100ฯ=
20ฯโ
62.83
51โร100ฯ=20ฯ
8
x
)
+
(y2โ
10y)=
โ16
Step 2: Complete the square for x:
Half of 8 = 4, square it = 16
(x2+8x+16)
Step 3: Complete the square for y:
Half of โ10 = โ5, square it = 25
(y2โ10y+25)
Step 4: Add the same values to the right side:
(x2+8x+16)+(y2โ10y+25)=โ16+16+25(x+4)2+(yโ5)2=25