HiSET - Area | Practice Hub

Find the area of a circle with the following radius:

$36\pi$

$6\pi$

$12\pi$

$66\pi$

$18\pi$

Explanation

The area of a circle is found using the following formula:

$A=\pi r^2$

In this formula the variable, , is the radius. Let's substitute in our known values and solve for the area.

$A=\pi 6^2$

$A=36\pi$

Find the area of a square with the following side length:

Explanation

We can find the area of a circle using the following formula:

In this equation the variable, , represents the length of a single side.

Substitute and solve.

A farmer has to make a square pen to hold chickens. If each chicken has to have $1 ;\textup{m}^2$ of area to roam and there are chickens total, what is the length of the amount of fencing required to pen in the chickens?

$20;\textup{m}$

$25 ;\textup{m}$

$50 ;\textup{m}$

$75 ;\textup{m}$

$83;\textup{m}$

Explanation

A square has area formula

The total area required for the chickens will be

$25 ;\textup{m}^2$

since each chicken requires $1;\textup{m}^2$ of space and there are chickens.

Thus, we have

for the length of our chicken fence.

$\sqrt{25}=\sqrt{l^2}^$

Since there are four sides of a square and each side has a length of 5, the total length of fence required is

$5\times4 = 20;\textup{m}$ .

Give the area of the circle on the coordinate plane with equation

$x^{2}+y^{2}+ 6x-20y+ 9=0$ .

$100 \pi$

$50 \pi$

$10 \pi$

$20 \pi$

$40 \pi$

Explanation

We must first rewrite the equation of the circle in standard form

$(x-h)^{2} + (y-k)^{2}= r^{2}$ ;

will be the radius.

$x^{2}+y^{2}+ 6x-20y+ 9=0$

Subtract 9 from both sides, and rearrange the terms remaining on the left as follows:

$x^{2}+y^{2}+ 6x-20y+ 9- 9 =0- 9$

$x^{2}+ 6x + \underline{; ; ; ; ; ; }+y^{2}-20y+ \underline{; ; ; ; ; ; } = - 9$

Note that blanks have been inserted after the linear terms. In these blanks, complete two perfect square trinomials by dividing linear coefficients by 2 and squaring:

$(6 \div 2)^{2} = 3^{2} = 9$

$(20 \div 2)^{2} = 10^{2} = 100$

Add to both sides, as follows:

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = - 9+9+ 100$

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = 100$

Rewrite the expression on the left as the sum of the squares of two binomials.

$(x+3)^{2} + (y-10)^{2} = 100$

The equation is now in standard form.

$r^{2} = 100$

The area of this circle can be found using this formula:

$A = \pi r^{2} = \pi (100) = 100 \pi$ ,

the correct response.

A circle on the coordinate plane has center and passes through . Give its area.

$100 \pi$

$10 \pi$

$20 \pi$

$25 \pi$

The information given is insufficient to answer the question

Explanation

The radius of the circle is the distance between its center and a point that it passes through. This radius can be calculated using the distance formula:

$r = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$ .

Setting $x_{1} = 2, x_{2} = 10, y_{1}=9, y_{2} =3$ :

$r = \sqrt{(10-2)^{2}+(3-9)^{2} }$

$= \sqrt{8^{2}+(-6)^{2} }$

$= \sqrt{64+36}$

$= \sqrt{100}$

The area of a circle can be calculated by substituting 10 for in the equation

$A = \pi r^{2}$

$= \pi (10)^{2}$

$= 100 \pi$ ,

the correct response.

A cube has surface area 6. Give the surface area of the sphere that is inscribed inside it.

$\pi$

$\frac{1}{3} \pi$

$\frac{2}{3} \pi$

$\frac{1}{6} \pi$

$\frac{1}{2} \pi$

Explanation

A cube with surface area 6 has six faces,each with area 1. As a result, each edge of the cube has length the square root of this, which is 1.

This is the diameter of the sphere inscribed in the cube, so the radius of the sphere is half this, or $r= \frac{1}{2}$ . Substitute this for in the formula for the surface area of a sphere:

$A = 4\pi r^{2} = 4\pi \left ( \frac{1}{2} \right )^{2} = 4\pi \left ( \frac{1}{4} \right ) = \pi$ ,

the correct choice.

Census

Source: United States Census Bureau

The average population density of a geographic area is defined to be the average number of residents per square mile.

Above is a table with the land areas and populations of five states.Which state among the five has the greatest population density?

Tennessee

Alabama

Arkansas

Kentucky

Mississippi

Explanation

For each state, divide the population by the land area. We can round each figure to the nearest whole for simplicity's sake.

Alabama:

$4,860,000 \div 50,645 \approx 96$ persons per square mile.

Arkansas:

$2,980,000 \div 52,035 \approx 57$ persons per square mile.

Kentucky:

$4,430,000 \div 39,486 \approx 112$ persons per square mile.

Mississippi:

$2,990,000 \div 46,923 \approx 64$ persons per square mile.

Tennessee:

$6,600,000 \div 41,235 \approx 160$ persons per square mile.

Tennessee has the greatest population density among the five states.

The perimeter of a square is . In terms of , give the area of the square.

$\frac{1}{16}P^{2}$

$\frac{1}{4}P^{2}$

$\frac{1}{16} \sqrt{P}$

$\frac{1}{4} \sqrt{P}$

Explanation

Since a square comprises four segments of the same length, the length of one side is equal to one fourth of the perimeter of the square, which is $\frac{1}{4}P$ . The area of the square is equal to the square of this sidelength, or

$\left (\frac{1}{4}P \right )^{2} = \left (\frac{1}{4} \right )^{2} P^{2} = \frac{1}{16}P^{2}$ .

The volume of a sphere is equal to $108 \pi$ . Give the surface area of the sphere.

$36 \pi \sqrt[3]{9}$

$36 \pi \sqrt[3]{3}$

$12 \pi \sqrt[3]{3}$

$12 \pi \sqrt[3]{9}$

None of the other choices gives the correct response.

Explanation

The volume of a sphere can be calculated using the formula

$V= \frac{4}{3} \pi r^{3}$

Solving for :

Set $V = 108 \pi$ . Multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 108 \pi$

$\pi r^{3} = 81\pi$

Divide by $\pi$ :

$\frac{\pi r^{3} }{\pi}= \frac{81\pi}{\pi}$

$r^{3} = 81$

Take the cube root of both sides:

$r = \sqrt[3]{81} = \sqrt[3]{27(3)} = \sqrt[3]{27} \cdot \sqrt[3]{3} = 3 \sqrt[3]{3}$

Now substitute for in the surface area formula:

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

$A = 4 \pi (3 \sqrt[3]{3})^{2}= 4 \pi (3 )^{2}( \sqrt[3]{3})^{2} = 4 \pi (9) ( \sqrt[3]{3^{2}}) = 36 \pi \sqrt[3]{9}$ ,

the correct response.

County 1

The average population density of a geographic area is defined to be the average number of residents per square mile.

Above is the map of a county whose population is about 120,000. Which of the following is the best estimate of the average population density?

100 persons per square mile

50 persons per square mile

150 persons per square mile

200 persons per square mile

250 persons per square mile

Explanation

The county is in the shape of a trapezoid with bases of length and , and with height . Its area in square miles can be found by substituting in the formula for the area of a trapezoid: