Properties of polygons and circles - HiSET

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 . The area of the square is equal to the square of this sidelength, or

.

The volume of a sphere can be calculated using the formula

Solving for :

Set . Multiply both sides by :

Divide by :

Take the cube root of both sides:

Now substitute for in the surface area formula:

,

the correct response.

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

Square this sidelength to get the area of the plot:

,

the correct response.

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

Divide each term by 4:

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

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 . Substitute this for in the formula for the surface area of a sphere:

,

the correct choice.

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

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

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

;

will be the radius.

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

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:

Add to both sides, as follows:

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

The equation is now in standard form.

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

,

the correct response.

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:

.

Setting :

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

,

the correct response.

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

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

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

;

will be the radius.

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

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:

Add to both sides, as follows:

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

The equation is now in standard form.

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

,

the correct response.

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:

.

Setting :

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

,

the correct response.

Perimeters can be calculated using the following formula.

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

Substitute and solve.

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 yards - the correct response.

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 . Since a square comprises four sides of equal length, the perimeter is equal to four times this length, or .

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:

,

the correct response.

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

The perimeter of Trapezoid can be seen to be .

Perimeters can be calculated using the following formula.

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

Substitute and solve.

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 yards - the correct response.

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 . Since a square comprises four sides of equal length, the perimeter is equal to four times this length, or .