GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 21 Sep 2018, 01:11

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Math: Circles

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 49276

### Show Tags

Updated on: 25 Jan 2015, 09:32
68
74
CIRCLES

This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

--------------------------------------------------------
Get The Official GMAT Club's App - GMAT TOOLKIT 2.
The only app you need to get 700+ score!

[iOS App] [Android App]

--------------------------------------------------------

Definition

A line forming a closed loop, every point on which is a fixed distance from a center point. Circle could also be defined as the set of all points equidistant from the center.

Center -a point inside the circle. All points on the circle are equidistant (same distance) from the center point.

Radius - the distance from the center to any point on the circle. It is half the diameter.

Diameter -t he distance across the circle. The length of any chord passing through the center. It is twice the radius.

Circumference - the distance around the circle.

Area - strictly speaking a circle is a line, and so has no area. What is usually meant is the area of the region enclosed by the circle.

Chord - line segment linking any two points on a circle.

Tangent -a line passing a circle and touching it at just one point.
The tangent line is always at the 90 degree angle (perpendicular) to the radius of a circle.

Secant A line that intersects a circle at two points.

$$\pi$$ In any circle, if you divide the circumference (distance around the circle) by it's diameter (distance across the circle), you always get the same number. This number is called Pi and is approximately 3.142.

• A circle is the shape with the largest area for a given length of perimeter (has the highest area to length ratio when compared to other geometric figures such as triangles or rectangles)
• All circles are similar
• To form a unique circle, it needs to have 3 points which are not on the same line.

Circumference, Perimeter of a circle

Given a radius $$r$$ of a circle, the circumference can be calculated using the formula: $$Circumference=2\pi{r}$$

If you know the diameter $$D$$ of a circle, the circumference can be found using the formula: $$Circumference=\pi{D}$$

If you know the area $$A$$ of a circle, the circumference can be found using the formula: $$Circumference=\sqrt{4\pi{A}}$$

Area enclosed by a circle

Given the radius $$r$$ of a circle, the area can be calculated using the formula: $$Area={\pi}r^2$$

If you know the diameter $$D$$ of a circle, the area can be found using the formula: $$Area=\frac{{\pi}D^2}{4}$$

If you know the circumference $$C$$ of a circle, the area can be found using the formula: $$Area=\frac{C^2}{4{\pi}}$$

Semicircle

Half a circle. A closed shape consisting of half a circle and a diameter of that circle.

• The area of a semicircle is half the area of the circle from which it is made: $$Area=\frac{{\pi}r^2}{2}$$

• The perimeter of a semicircle is not half the perimeter of a circle. From the figure above, you can see that the perimeter is the curved part, which is half the circle, plus the diameter line across the bottom. So, the formula for the perimeter of a semicircle is: $$Perimeter=\pi{r}+2r=r(\pi+2)$$

• The angle inscribed in a semicircle is always 90°.

• Any diameter of a circle subtends a right angle to any point on the circle. No matter where the point is, the triangle formed with diameter is always a right triangle.

Chord

A line that links two points on a circle or curve.

• A diameter is a chord that contains the center of the circle.
• Below is a formula for the length of a chord if you know the radius and the perpendicular distance from the chord to the circle center. This is a simple application of Pythagoras' Theorem.
$$Length=2\sqrt{r^2-d^2}$$, where $$r$$ is the radius of the circle, $$d$$ is the perpendicular distance from the chord to the circle center.
• In a circle, a radius perpendicular to a chord bisects the chord. Converse: In a circle, a radius that bisects a chord is perpendicular to the chord, or In a circle, the perpendicular bisector of a chord passes through the center of the circle.

Angles in a circle

An inscribed angle is an angle ABC formed by points A, B, and C on the circle's circumference.

• Given two points A and C, lines from them to a third point B form the inscribed angle ∠ABC. Notice that the inscribed angle is constant. It only depends on the position of A and C.
• If you know the length $$L$$ of the minor arc and radius, the inscribed angle is: $$Angle=\frac{90L}{\pi{r}}$$

A central angle is an angle AOC with endpoints A and C located on a circle's circumference and vertex O located at the circle's center. A central angle in a circle determines an arc AC.

The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle.

• An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Arcs and Sectors

A portion of the circumference of a circle.

Major and Minor Arcs Given two points on a circle, the minor arc is the shortest arc linking them. The major arc is the longest. On the GMAT, we usually assume the minor (shortest) arc.

Arc Length The formula the arc measure is: $$L=2\pi{r}\frac{C}{360}$$, where C is the central angle of the arc in degrees. Recall that $$2\pi{r}$$ is the circumference of the whole circle, so the formula simply reduces this by the ratio of the arc angle to a full angle (360). By transposing the above formula, you solve for the radius, central angle, or arc length if you know any two of them.

Sector is the area enclosed by two radii of a circle and their intercepted arc. A pie-shaped part of a circle.

Area of a sector is given by the formula: $$Area=\pi{r^2}\frac{C}{360}$$, where: C is the central angle in degrees. What this formula is doing is taking the area of the whole circle, and then taking a fraction of that depending on the central angle of the sector. So for example, if the central angle was 90°, then the sector would have an area equal to one quarter of the whole circle.

Power of a Point Theorem

Given circle O, point P not on the circle, and a line through P intersecting the circle in two points. The product of the length from P to the first point of intersection and the length from P to the second point of intersection is constant for any choice of a line through P that intersects the circle. This constant is called the "power of point P".

If P is outside the circle:

$$PA*PD=PC*PB=Constant$$ - This becomes the theorem we know as the theorem of intersecting secants.

If P is inside the circle:

$$PA*PD=PC*PB=Constant$$ - This becomes the theorem we know as the theorem of intersecting chords.

Tangent-Secant

Should one of the lines be tangent to the circle, point A will coincide with point D, and the theorem still applies:

$$PA*PD=PC*PB=Constant$$

$$PA^2=PC*PB=Constant$$ - This becomes the theorem we know as the theorem of secant-tangent theorem.

Two tangents

Should both of the lines be tangents to the circle, point A coincides with point D, point C coincides with point B, and the theorem still applies:

$$PA*PD=PC*PB=Constant$$

$$PA^2=PC^2$$

$$PA=PC$$

Official GMAC Books:

The Official Guide, 12th Edition: DT #36; PS #33; PS #160; PS #197; PS #212; DS #42; DS #96; DS #114; DS #117; DS #160; DS #173;
The Official Guide, Quantitative 2th Edition: PS #33; PS #141; PS #145; PS #153; PS #162; DS #22; DS #58; DS #59; DS #95; DS #99;
The Official Guide, 11th Edition: DT #36; PS #30; PS #42; PS #100; PS #160; PS #206; PS #229; DS #23; DS #76; DS #86; DS #136; DS #152;

Generated from [GMAT ToolKit]

--------------------------------------------------------
Get The Official GMAT Club's App - GMAT TOOLKIT 2.
The only app you need to get 700+ score!

[iOS App] [Android App]

--------------------------------------------------------

Spoiler: :: Images
Attachment:

Math_cir_3.png [ 7.27 KiB | Viewed 131748 times ]
Attachment:

Math_cir_13.png [ 10.75 KiB | Viewed 131235 times ]
Attachment:

Math_cir_12.png [ 11.87 KiB | Viewed 131306 times ]
Attachment:

Math_cir_10.png [ 13.63 KiB | Viewed 131332 times ]
Attachment:

Math_cir_9.png [ 12.73 KiB | Viewed 131299 times ]
Attachment:

Math_cir_8.png [ 17.38 KiB | Viewed 160159 times ]
Attachment:

Math_cir_7.png [ 11.13 KiB | Viewed 131969 times ]
Attachment:

Math_cir_6.png [ 13.89 KiB | Viewed 131216 times ]
Attachment:

Math_cir_5.png [ 13.05 KiB | Viewed 131431 times ]
Attachment:

Math_cir_4.png [ 8.76 KiB | Viewed 131430 times ]
Attachment:

Math_cir_2.png [ 18.58 KiB | Viewed 131752 times ]
Attachment:

Math_cir_1.png [ 13.1 KiB | Viewed 131925 times ]
Attachment:

Math_icon_circles.png [ 3.34 KiB | Viewed 130283 times ]
Attachment:

Math_cir_11.png [ 13.79 KiB | Viewed 130159 times ]

_________________

Originally posted by Bunuel on 14 Dec 2009, 07:31.
Last edited by walker on 25 Jan 2015, 09:32, edited 10 times in total.
Text Edit
Joined: 20 Aug 2009
Posts: 286
Location: Tbilisi, Georgia
Schools: Stanford (in), Tuck (WL), Wharton (ding), Cornell (in)

### Show Tags

04 Feb 2010, 03:31
2
Amazing work!

I wish there were something like this in verbal section
Intern
Joined: 15 Feb 2010
Posts: 46
Location: Tokyo

### Show Tags

15 Feb 2010, 18:52
1
great post!
Intern
Joined: 17 Nov 2009
Posts: 35
Schools: University of Toronto, Mcgill, Queens

### Show Tags

18 Feb 2010, 11:50
1
Excellent Info.

+1
_________________

--Action is the foundational key to all success.

Manager
Joined: 15 Dec 2009
Posts: 64

### Show Tags

09 Jun 2010, 13:34
1
• If you know the length of the minor arc and radius, the inscribed angle is: 90L/nr

Please correct me if i am wrong but i think the formula should be : 180L/nr
Math Expert
Joined: 02 Sep 2009
Posts: 49276

### Show Tags

09 Jun 2010, 13:56
6
chauhan2011 wrote:
• If you know the length of the minor arc and radius, the inscribed angle is: 90L/nr

Please correct me if i am wrong but i think the formula should be : 180L/nr

If you know the length $$L$$ of the minor arc and radius, the inscribed angle is: $$Inscribed \ Angle=\frac{90L}{\pi{r}}$$.

The way to derive the above formula:

Length of minor arc is $$L= \frac{Central \ Angle}{360}* Circumference$$ --> $$L= \frac{Central \ Angle}{360}* 2\pi{r}$$ --> $$L= \frac{Central \ Angle}{180}* 2\pi{r}$$ --> $$Central \ Angle=\frac{180L}{\pi{r}}$$ (so maybe you've mistaken central angle for inscribed angle?).

The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle: $$Central \ Angle=2*Inscribed \ Angle$$.

So, $$2*Inscribed \ Angle=\frac{180L}{\pi{r}}$$ --> $$Inscribed \ Angle=\frac{90L}{\pi{r}}$$.

Hope it helps.
_________________
Senior Manager
Joined: 25 Feb 2010
Posts: 375

### Show Tags

30 Jun 2010, 13:21
1
Waoooooo

Thanks a lot.
_________________

GGG (Gym / GMAT / Girl) -- Be Serious

Its your duty to post OA afterwards; some one must be waiting for that...

CEO
Joined: 17 Nov 2007
Posts: 3458
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40

### Show Tags

30 Jun 2010, 13:43
1
+1
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | Limited GMAT/GRE Math tutoring in Chicago

Senior Manager
Joined: 05 Jul 2010
Posts: 332

### Show Tags

07 Jul 2010, 16:41
2
Please note that the theorem of intersecting cords has a mistake. For math newbies, it could be dangerous. The same mistake is repeated in the iPhone App.

Posted from GMAT ToolKit
CEO
Joined: 17 Nov 2007
Posts: 3458
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40

### Show Tags

07 Jul 2010, 18:12
Abhicoolmax,

Thanks for reporting. I think there is a mistake in image only in GMAT ToolKit.
We are going to submit 1.3.7. update today and I will add the image from the thread.
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | Limited GMAT/GRE Math tutoring in Chicago

Manager
Joined: 16 Feb 2010
Posts: 185

### Show Tags

30 Aug 2010, 16:40
thanks, great post !
Manager
Joined: 15 Apr 2010
Posts: 123

### Show Tags

08 Sep 2010, 04:45
• An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Can anyone explain me the text in red?

Thanks
Senior Manager
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 367
Location: Milky way
Schools: ISB, Tepper - CMU, Chicago Booth, LSB

### Show Tags

16 Sep 2010, 23:29
2
tingle15 wrote:
• An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Can anyone explain me the text in red?

Thanks

The maximum central angle is 180 degrees and the maximum inscribed angle is 90 (exactly half of the corresponding central angle).

Angles inscribed on the arc are supplementary means that the angles adds to 180 degree. Two angles are supplementary if the sum of their angles equals 180 degree.

I found this URL useful -- http://www.mathopenref.com/arccentralangletheorem.html Have a look.
_________________

Support GMAT Club by putting a GMAT Club badge on your blog

Senior Manager
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 367
Location: Milky way
Schools: ISB, Tepper - CMU, Chicago Booth, LSB

### Show Tags

16 Sep 2010, 23:33
Now I am confused. Shouldn't the statement say that -- Angles inscribed on the arc are [highlight]complementary[/highlight] since the maximum value of the inscribed angle can be 90 degree only.
_________________

Support GMAT Club by putting a GMAT Club badge on your blog

Math Expert
Joined: 02 Sep 2009
Posts: 49276

### Show Tags

18 Sep 2010, 20:56
1
tingle15 wrote:
• An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Can anyone explain me the text in red?

Thanks

Supplementary angles are two angles that add up to 180°.

Complementary angles are two angles that add up to 90°.

Two angles inscribed on the arc (one from the one side and one from the another) are supplementary, they add up to 180° (on the diagram alpha and the angel opposite to alpha add up to 180°).

Hope it's clear.
_________________
Senior Manager
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 273
Schools: Chicago,Tuck,Oxford,cambridge

### Show Tags

20 Sep 2010, 10:05
hey thanx a lot bunuel for these circles!!as i needed bit more material for concepts...
Manager
Joined: 27 May 2008
Posts: 105

### Show Tags

07 Oct 2010, 20:50
Thanks for the concepts..
Manager
Joined: 01 Nov 2010
Posts: 134
Location: Zürich, Switzerland

### Show Tags

11 Nov 2010, 15:49
The post is just awesome....!! Thanks once again Bunuel...!
Manager
Joined: 26 Sep 2010
Posts: 138
Nationality: Indian
Concentration: Entrepreneurship, General Management

### Show Tags

12 Feb 2011, 21:31
Just came across this, great stuff!
_________________

You have to have a darkness...for the dawn to come.

Intern
Joined: 13 Oct 2010
Posts: 2

### Show Tags

27 Apr 2011, 18:51
Bunuel,

What angle would be the angle opposite to alpha? I am a bit confused.
Re: Math: Circles &nbs [#permalink] 27 Apr 2011, 18:51

Go to page    1   2   3    Next  [ 43 posts ]

Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.