ISEE Upper Level Quantitative Reasoning - How to find the area of a circle

Find the area of a circle with a radius of 6in.

$36\pi \text{ in}^2$

$12\pi \text{ in}^2$

$18\pi \text{ in}^2$

$24\pi \text{ in}^2$

$42\pi \text{ in}^2$

Explanation

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

Now, we know the radius of the circle is 6in.

Knowing this, we can substitute into the formula. We get

$A = \pi \cdot (6\text{in})^2$

$A = \pi \cdot 36\text{in}^2$

$A = 36\pi \text{ in}^2$

Target

The above figure depicts a dartboard, in which .

A blindfolded man throws a dart at the target. Disregarding any skill factor and assuming he hits the target, what are the odds against his hitting the white inner circle?

15 to 1

16 to 1

7 to 1

8 to 1

Explanation

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The inner and outer circles have radii 1 and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$ - this is the white inner circle.

$A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$

The area of the portion of the target outside the white inner circle is $16 \pi - \pi = 15 \pi$ , so the odds against hitting the inner circle are

$\frac{15 \pi}{\pi} = \frac{15}{1}$ - that is, 15 to 1 odds against.

Let $\pi = 3.14$ .

Find the area of a circle with a diameter of 14cm. If necessary, round to the nearest tenths.

$153.9\text{cm}^2$

$113.1\text{cm}^2$

$201.0\text{cm}^2$

$150.7\text{cm}^2$

$153.8\text{cm}^2$

Explanation

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the diameter of the circle is 14cm. We also know that the diameter is two times the radius. Therefore, the radius is 7cm.

We also know $\pi = 3.14$ .

Knowing all of this, we can substitute into the formula. We get

$A = 3.14 \cdot (7\text{cm})^2$

$A = 3.14 \cdot 49\text{cm}^2$

$A = 153.86\text{cm}^2$

We were asked to round to the nearest tenth. So, we get

$A = 153.9\text{cm}^2$

You have a circular window in your vacation room. It has a radius of 9 inches. What is the area of the window?

$81 \pi in^2$

$18 \pi in^2$

$36 \pi in^2$

$81 \pi in$

Explanation

You have a circular window in your vacation room. It has a radius of 9 inches. What is the area of the window?

To find the area of a circle, use the following formula:

$A=\pi r^2$

Now, we know the radius, so we just need to plug it in and solve.

$A=\pi (9 in)^2=81 \pi in^2$

So, our answer:

$81 \pi in^2$

The radius of a circle is $\frac{2}{\sqrt{\pi }}$ . Give the area of the circle.

$4\pi$

$\frac{4}{\pi }$

$2\pi$

Explanation

The area of a circle can be calculated as $Area=\pi r^2$ , where is the radius of the circle, and $\pi$ is approximately .

$Area=\pi r^2=\pi\times (\frac{2}{\sqrt{\pi}})^2=\pi\times \frac{4}{\pi}\Rightarrow Area=4$

Find the area of a circle with a diameter of 14cm.

$49\pi \text{ cm}^2$

$196\pi \text{ cm}^2$

$21\pi \text{ cm}^2$

$63\pi \text{ cm}^2$

$56\pi \text{ cm}^2$

Explanation

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the diameter of the circle is 14cm. We also know the diameter is two times the radius. Therefore, the radius is 7cm.

Knowing this, we can substitute into the formula. We get

$A = \pi \cdot (7\text{cm})^2$

$A = \pi \cdot 49\text{cm}^2$

$A = 49\pi \text{ cm}^2$

You are conducting fieldwork, when you find a tree whose radius at chest height is . What is the area of a cross section of the tree at chest height?

$A=2.25\pi m^2$

$A=3\pi m^2$

$A=4.50\pi m^2$

Explanation

You are conducting fieldwork, when you find a tree whose radius at chest height is . What is the area of a cross section of the tree at chest height?

Begin with the formula for area of a circle

$A=\pi r^2$

Now, we know r, so all we have to do is plug it in and solve.

$A=\pi (1.5m)^2=\pi 2.25m^2=2.25\pi m^2$

So, our answer is:

$A=2.25\pi m^2$

Target

In the above figure, .

What percent of the figure is shaded gray?

$62 \frac{1}{2} %$

$66 \frac{2}{3} %$

Explanation

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$