Area of Polygons - SSAT Upper Level Quantitative

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. In this problem we have:

or

So the area of the parallelogram would be:

The area of a square is given by:

weher is the side length of a square. In this problem we have , so we can write:

Then:

or:

Solution 1:

We can divide the rectangle width and height by the square side length and multiply the results:

rectangle width square length =

rectangle heightsquare length =

Solution 2:

As the results of the division of rectangle width and height by the square length are integers and do not have a residual, we can say that the squares can be perfectly fitted in the rectangle. Now in order to find the number of squares we can divide the rectangle area by the square area:

Rectangle area = square inches

Square area = square inches

So we can get:

The area of a rectangle is given by multiplying the width times the height. As a formula:

Where:

is the width and is the height. So we can get:

The area of a rectangle is given by multiplying the width times the height. That means:

where:

width and height.

We know that: . Substitube the in the area formula:

Now we should solve the equation for :

The equation has two answers, one positive and one negative . As the length is always positive, the correct answer is inches.

A rectangle has two congruent diagonals. A diagonal of a rectangle divides it into two identical right triangles. The diagonal of the rectangle is the hypotenuse of these triangles. We can use the Pythagorean Theorem to find the length of the diagonal if we know the width and height of the rectangle.

where:

is the width of the rectangle
is the height of the rectangle

First, we find the height of the rectangle:

So we can write:

inches

As a rectangle has two diagonals with the same length, the sum of the diagonals is inches.

First we need to find the height of the rectangle. Since the width and the diagonal lengths are known, we can use the Pythagorean Theorem to find the height of the rectangle:

So we have:

So we can get:

Let be the length of the rectangle. Then its width is 60% of this, or . The perimeter is the sum of the lengths of its sides, or

; we set this equal to 800 inches and solve for :

The width is therefore

.

The product of the length and width is the area:

square inches.

Let be the length of the rectangle. Then its width is three-fourths of this, or . The perimeter is the sum of the lengths of its sides, or

.

Set this equal to 490 centimeters and solve for :

The length of the rectangle is 140 centimeters; the width is three-fourths of this, or

centimeters.

The area is the product of the length and the width:

square centimeters.

Let be the length in feet. Then the width of the rectangle in feet is four-sevenths of this, or . The area is equal to the product of the length and the width, so set up this equation and solve for :

Since this is the length in feet, we multiply this by 12 to get the length in inches:

The area of a rectangle is the product of the length and its height, Rectangle A has area square inches; Rectangle B has area square inches.

The area of Rectangle C is the mean of these areas, or

square inches, so its height is this area divided by its length:

inches.

The area of a rectangle is the product of the length and its height.

Rectangle A has area square inches, and Rectangle B has an area of square inches, so the sum of their areas is square inches. This is the area of Rectangle C; divide it by height inches to get a length of

inches. This answer is not among the given choices.

We begin this problem by finding the difference of two areas: the larger rectangle bounded by the outer edge of the border and the smaller rectangle that is the court itself.

The larger rectangle is square feet , and the court is square feet .

The difference, , repesents the area of the border.

Now we divide this by , which is just a bit over . But since we can't leave square feet unpainted, we have to round up to cans of paint.

A rhombus, by definition, has four sides of equal length. Therefore, , and, by the Multiplication Property, . Also, since and are the midpoints of their respective sides, and . Combining these statements, and letting :

Also, both and are altitudes of the rhombus; they are congruent, and we will call their common length (height).

The figure, with the lengths, is below.

The area of the entire Rhombus is the product of its height and the length of a base , so

.

Rectangle has as its length and width and , so its area is their product , Since

,

From the Division Property, it follows that

,

and

.

This makes 450 the area of Rectangle .

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces):

The total area:

Since a square has four sides of equal length, the solid looks like this:

The areas of each of the individual surfaces, each of which is a rectangle, are the product of their dimensions:

Front, back, top, bottom (four surfaces):

Left, right (two surfaces):

The total surface area is therefore

The area of Mark's lawn is . The amount of grass seed he needs is pounds.

He has two options.

Option 1: he can buy three fifty-pound bags for

Option 2: he can buy two fifty-pound bags and four ten-pound bags for

The first option is the more economical.

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. In this problem we have:

or

So the area of the parallelogram would be:

The area of a square is given by:

weher is the side length of a square. In this problem we have , so we can write:

Then:

or:

Solution 1:

We can divide the rectangle width and height by the square side length and multiply the results:

rectangle width square length =

rectangle heightsquare length =

Solution 2:

As the results of the division of rectangle width and height by the square length are integers and do not have a residual, we can say that the squares can be perfectly fitted in the rectangle. Now in order to find the number of squares we can divide the rectangle area by the square area:

Rectangle area = square inches

Square area = square inches

So we can get:

The area of a rectangle is given by multiplying the width times the height. As a formula:

Where:

is the width and is the height. So we can get: