Properties of polygons and circles - HiSET

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Question

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

Answer

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.

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Question

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

Answer

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

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

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Question

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

Answer

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.

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Question

Express the area of a square plot of land 60 feet in sidelength in square yards.

Answer

One yard is equal to three feet, so convert 60 feet to yards by dividing by conversion factor 3:

$60 \textup{ ft} \div 3 \textup{ ft / yd}= 20 \textup{ yd}$

Square this sidelength to get the area of the plot:

$20 \textup{ yd} \times 20 \textup{ yd} = 400 \textup{ yd} ^{2}$ ,

the correct response.

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Question

A square has perimeter . Give its area in terms of .

Answer

Divide the perimeter to get the length of one side of the square.

$s= \frac{P}{4}$

$s= \frac{12y+4}{4}$

Divide each term by 4:

$s= \frac{12y}{4}+ \frac{4}{4} = 3y+ 1$

Square this sidelength to get the area of the square. The binomial can be squared by using the square of a binomial pattern:

$A= s^{2}$

$=( 3y+ 1)^{2}$

$=( 3y )^{2}+ 2 (3y)(1)+ 1^{2}$

$= 3^{2}y ^{2}+ 2 (3 )(1)y+ 1^{2}$

$=9y ^{2}+ 6y+ 1$

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Question

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

Answer

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.

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Question

Find the area of a circle with the following radius:

Answer

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.

$A=\pi 6^2$

$A=36\pi$

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Question

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

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

Answer

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.

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Question

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

Answer

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.

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Question

Find the area of a circle with the following radius:

Answer

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.

$A=\pi 6^2$

$A=36\pi$

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Question

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

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

Answer

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.

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Question

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

Answer

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.

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Question

A five sided irregular polygon has sides of the following lengths:

$1,\ 4,\ 7,\ 8,\ \textup{and } 13$

Find its perimeter.

Answer

Perimeters can be calculated using the following formula.

In this formula, the variable, , represents a side of the polygon.

Substitute and solve.

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Question

Give the perimeter of a regular octagon in yards if the length of each side is feet.

Answer

The perimeter of a regular octagon - the sum of the lengths of its (eight congruent) sides - is eight times the common sidelength, so the perimeter of the octagon is feet. One yard is equivalent to three feet, so divide this by conversion factor 3 to get $\frac{8x}{3}$ yards - the correct response.

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Question

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

Answer

The length of one side of a square is equal to the square root of its area, so, if the area of a square is , the common sidelength is $\sqrt{A}$ . Since a square comprises four sides of equal length, the perimeter is equal to four times this length, or $4\sqrt{A}$ .

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Question

The perimeter of a regular octagon is $16\sqrt{6}$ . Give the length of one side.

Answer

A regular octagon has eight sides of equal length. Its perimeter is equal to the sum of the lengths of its sides, so the length of one side can be computed by dividing the perimeter by 8, as follows:

$l = \frac{P}{8}= \frac{16\sqrt{6}}{8} = \frac{16}{8} \sqrt{6}= 2\sqrt{6}$ ,

the correct response.

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Question

Hexagon is regular. Give the perimeter of Trapezoid .

Answer

A regular hexagon can be divided into six equilateral triangles, as follows:

The perimeter of Trapezoid can be seen to be .

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Question

A five sided irregular polygon has sides of the following lengths:

$1,\ 4,\ 7,\ 8,\ \textup{and } 13$

Find its perimeter.

Answer

Perimeters can be calculated using the following formula.

In this formula, the variable, , represents a side of the polygon.

Substitute and solve.

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Question

Give the perimeter of a regular octagon in yards if the length of each side is feet.

Answer

The perimeter of a regular octagon - the sum of the lengths of its (eight congruent) sides - is eight times the common sidelength, so the perimeter of the octagon is feet. One yard is equivalent to three feet, so divide this by conversion factor 3 to get $\frac{8x}{3}$ yards - the correct response.

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Question

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

Answer

The length of one side of a square is equal to the square root of its area, so, if the area of a square is , the common sidelength is $\sqrt{A}$ . Since a square comprises four sides of equal length, the perimeter is equal to four times this length, or $4\sqrt{A}$ .

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