ISEE Upper Level

Circles

Q: How do I find the area of a half-circle?

Just find the area of the whole circle using `A = \pi \cdot r^2`, and then multiply your answer by `\frac{1}{2}` (which is the same as dividing by 2).

Circumference, area, arcs, sectors, central angles, and radius/diameter — excludes composite shapes involving circles (see area-perimeter-composite)

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Have you ever thought about how a pizza is just a giant, delicious math problem? 🍕 When you study circles for the ISEE, thinking about pizza makes it so much easier!

Let's break down the parts of a circle. The center is the very middle. The radius is a straight line from the center to the edge—think of it as the length of one perfect slice of pizza. The diameter goes all the way across the circle through the center. It's just two slices back-to-back, so the diameter is always twice as long as the radius!

Next, we have the circumference. This is the distance all the way around the outside of the circle. In pizza terms, it's the crust! Finally, the area is all the space inside the circle—that's where all the cheese and pepperoni go. 🧀

On the ISEE, you'll see circles in both the Quantitative Reasoning and Mathematics Achievement sections. The test makers love to check if you know the difference between the crust (circumference) and the cheese (area). They also use a special math symbol called $π$ (pi). It sounds like dessert, but it's actually a number that equals about $3.14$ . The best part? On the ISEE, you usually don't even have to do the messy math with decimals. You can just leave $π$ right in your answer! Let's grab a slice and learn the rules. 🚀

Key Formula

Area (the cheese):

A = π \cdot r^{2}

and Circumference (the crust):

C = π \cdot d

C = 2 \cdot π \cdot r

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

If the area of a circle is numerically equal to

\frac{3}{4}

of its circumference, what is the diameter of the circle?

\frac{3}{2}

3

6

9 π

Show Solution

Let $r$ be the radius of the circle.

The area of a circle is given by $A = π r^{2}$ .
The circumference of a circle is given by $C = 2 π r$ .
According to the problem, the area is numerically equal to $\frac{3}{4}$ of its circumference:
$π r^{2} = \frac{3}{4} (2 π r)$
$π r^{2} = \frac{3}{2} π r$
Since $r$ cannot be zero for a circle, we can divide both sides by $π r$ :
$r = \frac{3}{2}$
The diameter $d$ is twice the radius:
$d = 2 r = 2 \times \frac{3}{2} = 3$
Thus, the diameter of the circle is $3$ units.

Answer: B

Q2 Hard

A circle has a central angle of

12 0^{\circ}

. If the arc subtended by this angle has a length of

8 π

units, what is the area of the circle?

64 π

square units

72 π

square units

96 π

square units

144 π

square units

Show Solution

The formula for the length of an arc $L$ is $L = \frac{θ}{36 0 ^{\circ}} \times 2 π r$ , where $θ$ is the central angle in degrees and $r$ is the radius.

Given $θ = 12 0^{\circ}$ and $L = 8 π$ .
Substitute these values into the formula:
$8 π = \frac{120}{360} \times 2 π r$
Simplify the fraction:
$8 π = \frac{1}{3} \times 2 π r$
$8 π = \frac{2 π r}{3}$
To solve for $r$ , multiply both sides by $3$ and divide by $2 π$ :
$r = 8 π \times \frac{3}{2 π}$
$r = 4 \times 3$
$r = 12$ units.
Now, calculate the area of the circle using the formula $A = π r^{2}$ :
$A = π (1 2^{2})$
$A = 144 π$ square units.

Answer: D

Q3 Hard

A sector of a circle has an area of

15 π

square units and a radius of

6

units. What is the measure of the central angle of the sector in degrees?

9 0^{\circ}

12 0^{\circ}

15 0^{\circ}

18 0^{\circ}

Show Solution

The formula for the area of a sector $A_{sec t or}$ is $A_{sec t or} = \frac{θ}{36 0 ^{\circ}} \times π r^{2}$ , where $θ$ is the central angle in degrees and $r$ is the radius.

Given $A_{sec t or} = 15 π$ and $r = 6$ .
Substitute these values into the formula:
$15 π = \frac{θ}{360} \times π (6^{2})$
$15 π = \frac{θ}{360} \times 36 π$
Divide both sides by $π$ :
$15 = \frac{θ}{360} \times 36$
$15 = \frac{36 θ}{360}$
Simplify the fraction $\frac{36}{360}$ to $\frac{1}{10}$ :
$15 = \frac{θ}{10}$
Multiply both sides by $10$ to solve for $θ$ :
$θ = 15 \times 10$
$θ = 15 0^{\circ}$
Thus, the central angle of the sector is $15 0^{\circ}$ .

Answer: C

Q4 Hard

Two concentric circles have radii

r_{1}

and

r_{2}

, where

r_{2} > r_{1}

. The circumference of the inner circle is

12 π

units. The area of the region between the two circles (an annulus) is

45 π

square units. What is the circumference of the outer circle?

15 π

units

16 π

units

18 π

units

20 π

units

Show Solution

Let $r_{1}$ be the radius of the inner circle and $r_{2}$ be the radius of the outer circle.
1. Find $r_{1}$ from the circumference of the inner circle:

The circumference $C_{1} = 2 π r_{1}$ .
Given $C_{1} = 12 π$ .
$12 π = 2 π r_{1}$
Divide both sides by $2 π$ :
$r_{1} = 6$ units.
2. Find $r_{2}$ using the area of the annulus:

The area of an annulus is the area of the outer circle minus the area of the inner circle.
$A_{ann u l u s} = π r_{2}^{2} - π r_{1}^{2}$
Given $A_{ann u l u s} = 45 π$ .
$45 π = π r_{2}^{2} - π r_{1}^{2}$
Divide both sides by $π$ :
$45 = r_{2}^{2} - r_{1}^{2}$
Substitute $r_{1} = 6$ :
$45 = r_{2}^{2} - 6^{2}$
$45 = r_{2}^{2} - 36$
Add $36$ to both sides:
$r_{2}^{2} = 45 + 36$
$r_{2}^{2} = 81$
Take the square root (since radius must be positive):
$r_{2} = 9$ units.
3. Calculate the circumference of the outer circle:

The circumference $C_{2} = 2 π r_{2}$ .
$C_{2} = 2 π (9)$
$C_{2} = 18 π$ units.

Answer: C

Tips & Strategies

Memorize the difference between $π \cdot r^{2}$ and $2 \cdot π \cdot r$ . A cool trick: Area is measured in 'square' units, so its formula is the one with the little square ( $^{2}$ ) in it! 🧠
In Quantitative Reasoning, remember that $π$ is just a number (about $3.14$ ). If you need to estimate, just pretend $π$ is $3$ to quickly see which answer makes sense.
The ISEE doesn't penalize for guessing! If you're stuck on a circle problem and the answers have $π$ in them, eliminate the ones that look completely wrong and take your best guess.

Common Mistakes

Watch out for confusing the radius and diameter! Always double-check which one the question gives you. If they give you a diameter but you need area, you must cut it in half first: $r = \frac{d}{2}$ .
Don't forget what squaring means! For area, $7^{2}$ means $7 \cdot 7 = 49$ , NOT $7 \cdot 2 = 14$ . Test makers love to put $14 π$ as a trap answer!

Frequently Asked Questions

Do I need to multiply by 3.14 on the ISEE?

Usually, no! Most answers on the ISEE Mathematics Achievement section will just leave $π$ in the answer choice (like $14 π$ ). You'll only need to estimate with $3.14$ on comparison questions.

What is the Quantitative Comparison section?

It's a special type of puzzle on the ISEE where you compare Column A and Column B. You don't always have to find the exact number; you just need to figure out which side is bigger, or if they are perfectly equal!

How do I find the area of a half-circle?

Just find the area of the whole circle using $A = π \cdot r^{2}$ , and then multiply your answer by $\frac{1}{2}$ (which is the same as dividing by 2).

Is there a penalty for guessing wrong on the ISEE?

Nope! On the ISEE, you get points for right answers and zero points for wrong ones. Never leave a bubble blank on your answer sheet! ✏️

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