Area and Circumference of a Circle - SSAT Upper Level Quantitative

Card 1 of 216

0

Didn't Know

0

Didn't Know

Knew It

0

1 of 2019 left

Question

Find the area of a circle with a radius of 5.

Tap to reveal answer

Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the radius of the circle is 5, so plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 5^{2}=25\Pi$

← Didn't Know|Knew It →

Question

Find the area of a circle with a diameter of 6.

Tap to reveal answer

Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the diameter of the circle is 6, meaning we have to first find the radius. The diameter is twice the length of the radius of a circle, meaning to find radius divide the diameter by 2:

$\frac{6}{2}=3=r$

Now plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 3^{2}=9\Pi$

← Didn't Know|Knew It →

Question

Find the area of a circle with a diameter of 16.

Tap to reveal answer

Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the diameter of the circle is 16, meaning we have to first find the radius. The diameter is twice the length of the radius of a circle, meaning to find radius divide the diameter by 2:

$\frac{16}{2}=8=r$

Now plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 8^{2}=64\Pi$

← Didn't Know|Knew It →

Question

A circle on the coordinate plane has equation

$x^{2} + y^{2} = 75$ .

Which of the following gives the circumference of the circle?

Tap to reveal answer

Answer

The equation of a circle on the coordinate plane is

$x^{2} + y^{2} = r^{2}$ ,

where is the radius. Therefore,

$r ^{2} = 75$

and

$r = \sqrt{ 75} = \sqrt{25} \times \sqrt{ 3} = 5 \sqrt{ 3}$ .

The circumference of a circle is $2 \pi$ times is radius, which here would be

$C = 2 \pi \times 5 \sqrt{3} = 10\pi \sqrt{3}$ .

← Didn't Know|Knew It →

Question

A circle on the coordinate plane has equation

$x^{2} + y^{2} = 49$

Which of the following gives the circumference of the circle?

Tap to reveal answer

Answer

The equation of a circle on the coordinate plane is

$x^{2} + y^{2} = r^{2}$

where is the radius. Therefore,

$r^{2}= 49$

and

$r = \sqrt{49} = 7$ .

The circumference of a circle is $2 \pi$ times is radius, which here would be

$C= 2 \pi \cdot 7 = 14 \pi$

← Didn't Know|Knew It →

Question

Refer to the above diagram. Give the length of arc $\widehat{AB}$ .

Tap to reveal answer

Answer

The figure is a sector of a circle with radius 8; the sector has degree measure $180 ^{\circ } - 45 ^{\circ }= 135 ^{\circ }$ . The length of the arc is

$A = \frac{ 135^{\circ }}{360 ^{\circ }} \cdot 2\pi r$

$A = \frac{ 135^{\circ }}{360 ^{\circ }} \cdot 2\pi \cdot 8$

$A = \frac{3}{8} \cdot 16\pi$

$A = 6\pi$

← Didn't Know|Knew It →

Question

Find the area of a circle with a radius of 4.

Tap to reveal answer

Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the radius of the circle is 4, so plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 4^{2}=16\Pi$

← Didn't Know|Knew It →

Question

A $60 ^{\circ }$ central angle of a circle has a chord with length 9. Give the circumference of the circle.

Tap to reveal answer

Answer

The figure below shows $\angle AOB$ , which matches this description, along with its chord $\overline{AB}$ :

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved equilateral, so . This is the radius, so the circumference is

$C = 2 \pi r = 2 \pi \cdot 9 = 18 \pi$

← Didn't Know|Knew It →

Question

What is the area of a circle with a diameter of , rounded to the nearest whole number?

Tap to reveal answer

Answer

The formula for the area of a circle is

$\dpi{100} \pi r^{2}$

Find the radius by dividing 9 by 2:

$\dpi{100} \frac{9}{2}=4.5$

So the formula for area would now be:

$\dpi{100} \pi r^{2}=\pi (4.5)^{2}=20.25\pi \approx 63.6= 64$

← Didn't Know|Knew It →

Question

What is the area of a circle that has a diameter of inches?

Tap to reveal answer

Answer

The formula for finding the area of a circle is $\pi r^{2}$ . In this formula, represents the radius of the circle. Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius. In order to do this, we divide the diameter by .

$\frac{15}{2}=7.5$

Now we use for in our equation.

$\pi (7.5)^{2}=176.7146 : in^{2}$

← Didn't Know|Knew It →

Question

What is the area of a circle with a diameter equal to 6?

Tap to reveal answer

Answer

First, solve for radius:

$r=\frac{d}{2}=\frac{6}{2}=3$

Then, solve for area:

$A=r^2\pi=3^2\pi=9\pi$

← Didn't Know|Knew It →

Question

The diameter of a circle is $4\ cm$ . Give the area of the circle.

Tap to reveal answer

Answer

The area of a circle can be calculated using the formula:

$Area=\frac{\pi d^2}{4}$ ,

where is the diameter of the circle, and $\pi$ is approximately .

$Area=\frac{\pi d^2}{4}=\frac{\pi\times 4^2}{4}=4\pi \Rightarrow Area\approx 4\times 3.14\Rightarrow Area\approx 12.56 \ cm^2$

← Didn't Know|Knew It →

Question

The diameter of a circle is . Give the area of the circle in terms of .

Tap to reveal answer

Answer

The area of a circle can be calculated using the formula:

$Area=\frac{\pi d^2}{4}$ ,

where is the diameter of the circle and $\pi$ is approximately .

$Area=\frac{\pi (4t)^2}{4}=\frac{16\pi t^2}{4}=4\pi t^2 \Rightarrow Area\approx 4\times 3.14\times t^2$

$\Rightarrow Area\approx 12.56t^2$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

The circumference of a circle is inches. Find the area of the circle.

Let $\pi = 3.14$ .

Tap to reveal answer

Answer

First we need to find the radius of the circle. The circumference of a circle is $Circumference =2\pi r$ , where is the radius of the circle.

$12.56=2\times 3.14\times r\Rightarrow r=2\ in$

The area of a circle is $Area=\pi r^2$ where is the radius of the circle.

$Area=\pi r^2=3.14\times 2^2=12.56\ in^2$

← Didn't Know|Knew It →

Question

The perpendicular distance from the chord to the center of a circle is $\sqrt{2}t$ , and the chord length is $2\sqrt{2}t$ . Give the area of the circle in terms of .

Tap to reveal answer

Answer

Chord length = $2\sqrt{r^2-d^2}$ , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length = $2\sqrt{r^2-d^2}\Rightarrow (2\sqrt{2})t=2\sqrt{r^2-(\sqrt{2}t)^2}$

$\Rightarrow 8t^2=4(r^2-2t^2)\Rightarrow 4r^2=16t^2\Rightarrow r^2=4t^2\Rightarrow r=2t$

$Area=\pi r^2$ , where is the radius of the circle and $\pi$ is approximately .

$Area=\pi r^2=3.14\times (2t)^2=12.56t^2$

← Didn't Know|Knew It →

Question

A circle on the coordinate plane has equation

$x^{2} + y^{2} = 66$ .

Which of the following gives the area of the circle?

Tap to reveal answer

Answer

The equation of a circle on the coordinate plane is

$x^{2} + y^{2} = r^{2}$ ,

where is the radius. Therefore, in this equation,

$r^{2} = 66$ .

The area of a circle is found using the formula

$A = \pi r^{2}$ ,

so we substitute 66 for $r^{2}$ , yielding

$A = 66 \pi$ .

← Didn't Know|Knew It →

Question

Give the area of the above figure.

Tap to reveal answer

Answer

The figure is a semicircle - one-half of a circle - with radius 5.5, or $\frac{11}{2}$ . Its area is one-half of the square of the radius multiplied by $\pi$ - that is,

$A = \frac{1}{2} \pi r^{2}$

$A = \frac{1}{2} \pi \left (\frac{11}{2} \right )^{2}$

$A = \frac{1}{2} \pi \cdot \frac{121}{4}$

$A = \frac{121\pi}{8 }$

← Didn't Know|Knew It →

Question

Give the area of the figure in the above diagram.

Tap to reveal answer

Answer

The figure is a sector of a circle with radius 8; the sector has degree measure $180 ^{\circ } - 45 ^{\circ }= 135 ^{\circ }$ . The area of the sector is

$A = \frac{ 135^{\circ }}{360 ^{\circ }} \cdot \pi r^{2}$

$A = \frac{ 135^{\circ }}{360 ^{\circ }} \cdot \pi \cdot 8^{2}$

$A = \frac{ 3}{8} \cdot 64 \pi$

$A = 24 \pi$

← Didn't Know|Knew It →

Question

Find the area of a circle with a radius of 12.

Tap to reveal answer

Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the radius of the circle is 12, so plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 12^{2}=144\Pi$

← Didn't Know|Knew It →

0

Knew It