Area - SAT Subject Test in Math I

Card 0 of 20

Question

What is the area of the following kite?

Answer

The formula for the area of a kite:

$A = \frac{1}{2} (a)(b)$ ,

where represents the length of one diagonal and represents the length of the other diagonal.

Plugging in our values, we get:

$A = \frac{1}{2} (4: m)(5: m)$

$A = \frac{1}{2} (20: m^2) = 10: m^2$

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Question

Which of the following shapes is a kite?

Answer

A kite is a four-sided shape with straight sides that has two pairs of sides. Each pair of adjacent sides are equal in length. A square is also considered a kite.

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Question

The above figure depicts the rectangular swimming pool at an apartment. The apartment manager needs to purchase a tarp that will cover this pool completely. However, because of the cutting device the pool store uses, the length and the width of the tarp must each be a multiple of three yards. Also, the tarp must be at least one yard longer and one yard wider than the pool.

What will be the minimum area of the tarp the manager purchases?

Answer

Three feet make a yard, so the length and width of the pool are $\frac{50}{3} = 16 \frac{2}{3}$ yards and $\frac{35}{3} = 11 \frac{2}{3}$ yards, respectively. Since the dimensions of the tarp must exceed those of the pool by at least one yard, the tarp must be at least $17 \frac{2}{3}$ yards by $12 \frac{2}{3}$ yards; but since both dimensions must be multiples of three yards, we take the next multiple of three for each.

The tarp must be 18 yards by 15 yards, so the area of the tarp is the product of these dimensions, or

$18 \times 15 = 270$ square yards.

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Question

Note: Figure NOT drawn to scale.

Refer to the figure above, which shows a square inscribed inside a large triangle. What percent of the entire triangle has been shaded blue?

Answer

The shaded portion of the entire triangle is similar to the entire large triangle by the Angle-Angle postulate, so sides are in proportion. The short leg of the blue triangle has length 20; that of the large triangle, 30. Therefore, the similarity ratio is . The ratio of the areas is the square of this, or $3^{2}:2^{2}$ , or .

The blue triangle is therefore $\frac{4}{9}$ of the entire triangle, or $\frac{4}{9} \times 100 = 44 \frac{4}{9} %$ of it.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the area of $\Delta ABC$ ?

Answer

If we see hypotenuse $\overline{AC}$ as the base of the large triangle, then we can look at the segment perpendicular to it, $\overline{BD}$ , as its altitude. Therefore, the area of $\Delta ABC$ is

$A = \frac{1}{2} (AC) (BD)$

.

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, the square root of the product of the two:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{6 \cdot 24} = \sqrt{144} = 12$

The area of $\Delta ABC$ is therefore

$A = \frac{1}{2} (AC) (BD) = \frac{1}{2} \cdot 30 \cdot 12 = 180$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of $\Delta BDC$ to that of $\Delta ADB$ .

Answer

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{6 \cdot 24} = \sqrt{144} = 12$ .

The areas of $\Delta BDC$ and $\Delta ADB$ , each being right, are half the products of their legs, so:

The area of $\Delta ADB$ is $\frac{1}{2} \times (AD) (BD) = \frac{1}{2} \times 6 \times (BD)= 3(BD)$

The area of $\Delta BDC$ is $\frac{1}{2} \times (CD) (BD) = \frac{1}{2} \times 24 \times (BD)= 12(BD)$

The ratio of the areas is $\frac{12 (BD)}{3 (BD)} = \frac{12 }{3 } =\frac{4}{1}$ - that is, 4 to 1.

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Question

The above figure depicts the rectangular swimming pool at an apartment. The apartment manager needs to purchase a tarp that will cover this pool completely, but the store will only sell the material in multiples of ten square yards. How many square yards will the manager need to buy?

Answer

Three feet make a yard, so the length and width of the pool are $\frac{50}{3}$ yards and $\frac{35}{3}$ yards; the area of the pool, and that of the tarp needed to cover it, must be the product of these dimensions, or

$\frac{50}{3} \times \frac{35}{3}= \frac{1,750}{9} = 194 \frac{4}{9}$ square yards.

The manager will need to buy a number of square yards of tarp equal to the next highest multiple of ten, which is 200 square yards.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. In terms of area, $\Delta ADB$ is what fraction of $\Delta ABC$ ?

Answer

The area of $\Delta ADB$ , being right, is half the products of its legs, which is:

$\frac{1}{2} (AD) (BD) = \frac{1}{2} \times 6 (BD) = 3 (BD)$

The area of $\Delta ABC$ is one half the product of its base and height; we can use its hypotenuse as the base and as the height, so this area is

$\frac{1}{2} (AC) (BD) = \frac{1}{2} \times 30 (BD) = 15 (BD)$

Therefore, in terms of area, $\Delta ADB$ is $\frac{3 (BD)}{15 (BD)} = \frac{3}{15} = \frac{1}{5}$ of $\Delta ABC$ .

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Question

Note: figure NOT drawn to scale.

Refer to the above figure. Quadrilateral is a square. What is the area of Polygon ?

Answer

Polygon is a composite of $\Delta ABE$ and Square ; its area is the sum of the areas of the two figures.

$\Delta ABE$ is a right triangle; its area is half the product of its legs, which is

$A = \frac{1}{2} \cdot 5 \cdot 8 = 20$

$\overline{BE }$ is both one side of Square and the hypotenuse of $\Delta ABE$ ; its hypotenuse can be calculated from the lengths of the legs using the Pythagorean Theorem:

$BE = \sqrt{5^{2}+8^{2}} = \sqrt{25 + 64 } = \sqrt{89}$ .

Square has area the square of this, which is 89.

Polygon has as its area the sum of these two areas:

.

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Question

Find the area of a kite if the diagonal dimensions are and .

Answer

The area of the kite is given below. The FOIL method will need to be used to simplify the binomial.

$A=\frac{d1\cdot d2}{2}=\frac{(x+3)(x-6)}{2}= \frac{x^2-6x+3x-18}{2}$

$A= \frac{x^2-3x-18}{2}$

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Question

The diagonals of a kite are $\sqrt10:cm$ and $\sqrt5:cm$ . Find the area.

Answer

The formula for the area for a kite is

$A=\frac{d_1\cdot d_2}{2}$ , where and are the lengths of the kite's two diagonals. We are given the length of these diagonals in the problem, so we can substitute them into the formula and solve for the area:

$A = \frac{d1\cdot d2}{2} = \frac{\sqrt10 \cdot \sqrt5}{2} = \frac{\sqrt50}{2} = \frac{5\sqrt2}{2}:cm^2$

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Question

Find the area of a rhombus if the diagonals lengths are and .

Answer

Write the formula for the area of a rhombus:

$A=\frac{d_1 \cdot d_2}{2}$

Substitute the given lengths of the diagonals and solve:

$A=\frac{d1 \cdot d2}{2} = \frac{20:cm \cdot 40:cm}{2} =\frac{800:cm^2}{2} = 400:cm^2$

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Question

Find the area of a kite with diagonal lengths of and .

Answer

Write the formula for the area of a kite.

$A= \frac{d1\cdot d2}{2}$

Plug in the given diagonals.

$A= \frac{(a+b)\cdot (2a-2b)}{2}$

Pull out a common factor of two in and simplify.

$A= \frac{(a+b)\cdot2(a-b)}{2}$

Use the FOIL method to simplify.

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Question

Find the area of a rhombus if the diagonals lengths are and .

Answer

Write the formula for finding the area of a rhombus. Substitute the diagonals and evaluate.

$A=\frac{d1\cdot d2}{2}= \frac{2a \cdot 5a^2}{2}= 5a^3$

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Question

Find the area of a circle with a diameter of .

Answer

Write the formula for the area of a circle.

$A_{circle}= \pi r^2 = \frac{\pi d^2}{4}$

Substitute the diameter and solve.

$\frac{\pi (16)^2}{4} = \frac{\pi (16)(16)}{4}=\pi (16)(4)= 64\pi$

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Question

Figure not drawn to scale.

Find the area of the rectangle above when the perimeter is 36 in.

Answer

Because we know the perimeter is 36 inches, we can determine the length of side w based on the equation of the perimeter of a rectangle:

$\small Perimeter= 2(length)+2(width)$

$\small 36=2(12)+2w$

$\small 36=24+2w$

$\small 12=2w$

$\small w=6$

Side w is 6 in long.

Now that we know that side w is 6 inches long, we have everythinng we need to calculate the area of the rectangle.

$\small Area=(length)(width)$

$\small (12)(6)=72$

The area of the rectangle is 72 in2

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Question

Give the area of $\bigtriangleup ABC$ to the nearest whole square unit, where:

Answer

The area of a triangle, given its three sidelengths, can be calculated using Heron's formula:

$A = \sqrt{s(s-a)(s-b)(s-c)}$ ,

where , , and are the lengths of the sides, and $s = \frac{a+b+ c}{2}$ .

Setting, , , and ,

$s = \frac{13+17+25}{2} = \frac{55}{2} = 27.5$

and, substituting in Heron's formula:

$A = \sqrt{27.5(27.5-13)(27.5-17)(27.5-25 )}$

$= \sqrt{27.5(14.5)(10.5)(2.5 )}$

$= \sqrt{10,467.1875}$

$\approx 102$

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Question

Give the area of $\bigtriangleup ABC$ to the nearest whole square unit, where:

$m \angle A = 57^{\circ }$

Answer

The area of a triangle with two sides of lengths and and included angle of measure $\gamma$ can be calculated using the formula

$A = \frac{1}{2} ab \sin \gamma$ .

Setting , , and $\gamma = m \angle A = 57^{\circ }$ , then evaluating :

$A = \frac{1}{2} (19 )(25)\sin 57^{\circ }$

$A = \frac{1}{2} (19 )(25) (0.8387)$

$A \approx 199$ .

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Question

On the XY plane, line segment AB has endpoints (0, a) and (b, 0). If a square is drawn with segment AB as a side, in terms of a and b what is the area of the square?

Answer

Since the question is asking for area of the square with side length AB, recall the formula for the area of a square.

$A=\text{side}^2$

Now, use the distance formula to calculate the length of AB.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

let

$\(x_1,y_1)=(0,a) \(x_2,y_2)=(b,0)$

Now substitute the values into the distance formula.

$\d=\sqrt{(b-0)^2+(0-2)^2} \d=\sqrt{b^2+a^2}$

From here substitute the side length value into the area formula.

$\A=\text{side}^2 \A=\left(\sqrt{b^2+a^2}\right)^2 \A=b^2+a^2$

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Question

Find the area of a triangle with a height of $\frac{1}{2}$ and a base of $\frac{1}{4}$ .

Answer

Write the formula for the area of a triangle.

$A=\frac{1}{2}bh$

Substitute the base and height into the formula.

$A=\frac{1}{2}(\frac{1}{2})(\frac{1}{4})$

Simplify the fractions.

The answer is: $\frac{1}{16}$

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