SAT Math - Area | Practice Hub

Triangle

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?

$44\frac{4}{9} %$

$33 \frac{1}{3} %$

Insufficient information is given to answer the question.

Explanation

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.

Determine the area of a circle with a diameter of .

$9x^2 \pi$

$9x\pi$

$36x^2\pi$

$18x^2\pi$

$6x^2\pi$

Explanation

Write the formula for the area of a circle.

$A= \pi r^2$

The radius is half the diameter, .

Substitute the radius into the equation.

$A= \pi (3x)^2 = 9x^2 \pi$

The answer is: $9x^2 \pi$

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?

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

Cannot be determined

Explanation

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.

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

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

Explanation

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.

Find the area of a circle with a radius of .

$9\pi x^6$

$6\pi x^6$

$9\pi^2 x^6$

$6\pi^2 x^6$

$9\pi x^9$

Explanation

The area of a circle is $A=\pi r^2$ .

Substitute the radius and solve for the area.

$A=\pi (3x^3)^2 = \pi (3x^3)(3x^3) = 9\pi x^6$

The answer is: $9\pi x^6$

Find the area of a circle with a diameter of .

$64\pi$

$32\pi$

$16\pi$

$81\pi$

$256\pi$

Explanation

Write the formula for the area of a circle.

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

Substitute the diameter and solve.

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

Right_triangle

Note: Figure NOT drawn to scale.

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

Insufficient information is given to answer the question.

Explanation

, 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: