Quadrilaterals - ISEE Middle Level Math

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Question

Find the area of the following parallelogram:

Note: The formula for the area of a parallelogram is $A=b\times h$ .

Answer

The base of the parallelogram is 10, while the height is 5.

$A=b\times h$

$A=10\times5=50: in^2$

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Question

Find the area:

Answer

The area of a parallelogram can be determined using the following equation:

$\small A=bh$

Therefore,

$\small A=8\times3=24$

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Question

You can solve the area of a parallelogram when you know the lengths of each of the sides. True or False?

Answer

The area of a parallelogram is found by computing . In this situation, you would have the base which is the bottom side but you would not have the height measurement. Since you would not be able to solve for the area with just the side lengths, the statement is .

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Question

If a parallelogram has side lengths of and , what is the area?

Answer

To find the area of a parallelogram, you use the formula,

.

Since the height in this problem is not known, you cannot solve for area.

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Question

Find the area of a parallelogram with a base of 6 inches and a height of 9 inches.

Answer

To find the area of a parallelogram, we will use the following formula:

$\text{area of a parallelogram} = b \cdot h$

where b is the base and h is the height of the parallelogram.

Now, we know the base has a length of 6 inches. We also know it has a height of 9 inches. Knowing this, we can substitute into the formula. We get

$\text{area of a parallelogram} = 6\text{in} \cdot 9\text{in}$

$\text{area of a parallelogram} = 54\text{in}^2$

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Question

Find the area of the parallelogram with a base length of 6 and a height of 15.

Answer

Write the area formula of a parallelogram.

Substitute the dimensions into the formula.

The answer is:

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Question

What is the area of a rectangle with length $14\ cm$ and width $12\ cm$ ?

Answer

The formula for the area, , of a rectangle when we are given its length, , and width, , is .

To calculate this area, just multiply the two terms.

$A = 14\ cm * 12\ cm = 168; cm^{2}$

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Question

Order the following from least area to greatest area:

Figure A: A rectangle with length 10 inches and width 14 inches.

Figure B: A square with side length 1 foot.

Figure C: A triangle with base 16 inches and height 20 inches.

Answer

Figure A has area $10 \times 14 = 140$ square inches.

Figure B has area $12 \times 12 = 144$ square inches, 1 foot being equal to 12 inches.

Figure C has area $\frac{1}{2} \times 16 \times 20 = 160$ square inches.

The figures, arranged from least area to greatest, are A, B, C.

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Question

The area of the following rectangle is . Solve for .

Answer

The area of a rectangle can be found by multiplying the length by the width.

$A=l\times w$

$48=3x\times x=3x^2$

$\frac{48}{3}=\frac{3x^2}{3}$

$\sqrt{16}=\sqrt{x^2}$

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Question

Give the surface area of the above box in square inches.

Answer

Use the surface area formula, substituting :

$A = 2 \cdot 22 \cdot 12 + 2 \cdot 22 \cdot 12 + 2\cdot 12\cdot 12$

square inches

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Question

The following question is about the Jones family wanting to buy square foot tiles for their rectangular basement. Their basement perimeter is 74 feet, with one of the sides being 15 feet long.

How many square foot titles are the Jones family needing to purchase in order to tile their basement?

Answer

From the given information we know that the perimeter of the rectangular basement is 74 feet. We also know that one side of the rectangular basement is 15 feet. This means that the opposite side is also 15 feet long because the equivalent opposite sides rule of rectangles. In order to find the lengths of our other two sides of the rectangle, we need to subtract our two 15 feet sides from the perimeters 74 feet.

$74-30=44\ feet$ .

We know that the last two sides have to add up to 44 feet. Since the rules of rectangles say opposite sides are equivalent, we must take 44 feet and divide by the 2 sides. So 44 divided by 2 is 22 feet, meaning each side must be 22 feet. After adding up all the sides we can confirm that our perimeter is 74 feet.

Now we know all the sides of the rectangle, we are able to move to the next step, finding the area. We must find the area, because the tiles are square feet. So in order to find the area we must take the length of the rectangle and multiply it to the width.

$22\times 15= 330 ft^{2}$

Knowing the area of the rectangular basement we also know how many tile are needed to fill the basement for the Jones family. It is exactly 330 square feet tile needed.

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Question

The ratio of the perimeter of one square to that of another square is . What is the ratio of the area of the first square to that of the second square?

Answer

For the sake of simplicity, we will assume that the second square has sidelength 1; Then its perimeter is $4 \times 1 = 4$ , and its area is $1^{2} = 1$ .

The perimeter of the first square is $\frac{7}{4} \times 4 = 7$ , and its sidelength is $7 \div 4 = \frac{7}{4}$ . The area of this square is therefore $\left (\frac{7}{4} \right )^{2} =\frac{49}{16}$ .

The ratio of the areas is therefore $\frac{49}{16} : 1 = 49:16$ .

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Question

Give the area of the above rectangle in square feet.

Answer

Since 1 yard = 3 feet, multiply each dimension by 3 to convert from yards to feet:

$8 \textrm{ yd} \times 3 \textrm{ ft/yd} = 24 \textrm{ ft}$

$3 \textrm{ yd} \times 3 \textrm{ ft/yd} = 9 \textrm{ ft}$

Use the area formula, substituting :

$A = 24 \times 9 = 216$ square feet

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Question

Which of the following is equal to the area of a rectangle with length feet and width $1 \frac{1}{2}$ feet?

Answer

Multiply each dimension by 12 to convert from feet to inches:

$3 \times 12 = 36 \textrm{ in}$

$1 \frac{1}{2} \times 12 = 18 \textrm{ in}$

Now multiply this length and width to get the area:

$36 \times 18 = 648 \textrm{ in}^{2}$

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Question

A rectangular prism has length 24 inches, width 18 inches, and height 15 inches. Give its surface area in square feet.

Answer

Divide each dimension in inches by 12 to convert from inches to feet:

$L = 24 \div 12 = 2$ feet

$W = 18 \div 12 = 1 \frac{1}{2} = \frac{3}{2}$ feet

$H = 15 \div 12 = 1 \frac{1}{4} = \frac{5}{4}$ feet

Now, substitute in the surface area formula:

$A = 2\cdot 2 \cdot \frac{3}{2} + 2 \cdot \frac{3}{2}\cdot \frac{5}{4} + 2\cdot 2 \cdot \frac{5}{4}$

$A =6 + \frac{15}{4} + 5$

$A =6 + 3 \frac{3}{4} + 5$

$A =14\frac{3}{4}ft^{2}$

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Question

Which of the following is equal to the area of a rectangle with length meters and width meters?

Answer

Multiply each dimension by to convert meters to centimeters:

$4.3 \times 100 = 430$

$3.5 \times 100 = 350$

Multiply these dimensions to get the area of the rectangle in square centimeters:

$430 \times 350 = 150,500\textrm{ cm}^{2}$

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Question

Which of the following is equal to the area of a rectangle with length 27 inches and width 15 inches?

Answer

Divide each dimension by 12 to convert from inches to feet:

$27 \div 12 = 2.25$

$15 \div 12 = 1.25$

Multiply length by width:

$2.25 \times 1.25 = 2.8125$ square feet

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Question

Which of the following is equal to the area of a rectangle with length 250 centimeters and width 140 centimeters?

Answer

Divide each dimension by 100 to convert centimeters to meters:

$250 \div 100 = 2.5$

$140 \div 100 = 1.4$

Multiply length by width:

$2.5 \times 1.4 = 3.5$ square meters

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Question

A hectare is a unit of area equal to 10,000 square meters.

A rectangular plot of land measures 1,400 meters by 800 meters. Give the area of this plot in hectares.

Answer

Multiply length times width to get the area in square meters:

$A = 1,400 \times 800 = 1,120,000$ square meters

Divide by 10,000 to convert to hectares:

$1,120,000 \div 10,000 = 112$ hectares

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Question

The area of a rectangle with a length of and a width of is $84\ in^2$ . Find .

Answer

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

$7x\times 6x=84\Rightarrow 42x^2=84\Rightarrow x^2=2\Rightarrow x=\sqrt{2}\approx 1.41$

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