Quadrilaterals

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Questions 1 - 10

What is the area of the figure below?

Explanation

To find the area of the figure above, we need to slip the figure into two rectangles.

11.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 3$ $A=6\times 3$

To find our final answer, we need to add the areas together.

Annie has a piece of wallpaper that is by . How much of a wall can be covered by this piece of wallpaper?

Explanation

This problem asks us to calculate the amount of space that the wallpaper will cover. The amount of space that something covers can be described as its area. In this case area is calculated by using the formula $A=l \times w$

$A=3\times2$

In the figure, a square is inscribed in a circle. If the perimeter of the square is , then what is the area of the shaded region?

Explanation

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, let's find the area of the square.

From the given information, we can find the length of a side of the square.

$\text{Perimeter}=4(\text{side})$

$\text{side}=\frac{\text{Perimeter}}{4}$

Substitute in the value of the perimeter to find the length of a side of the square.

$\text{side}=\frac{8}{4}$

Simplify.

$\text{side}=2$

Now recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Substitute in the value of the side of the square to find the area.

$\text{Area of square}=2^2$

Simplify.

$\text{Area of square}=4$

Now, use the Pythagorean theorem to find the length of the diagonal of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}=\sqrt{2(\text{side})^2}$

Simplify.

$\text{Diagonal}=\text{side}\sqrt2$

Substitute in the value of the side of the square to find the length of the diagonal.

$\text{Diagonal}=2\sqrt2$

Recall that the diagonal of the square is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=2\sqrt2$

From the diameter, we can then find the radius of the circle:

$\text{Radius}=\frac{\text{Diameter}}{2}$

$\text{Radius}=\frac{2\sqrt2}{2}$

Simplify.

$\text{Radius}=\sqrt2$

Now, use the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of circle}=\pi\times (\sqrt2)^2$

Simplify.

$\text{Area of circle}=2\pi$

To find the area of the shaded region, subtract the area of the square from the area of the circle.

$\text{Area of Shaded Region}=\text{Area of circle}-\text{Area of square}$

$\text{Area of shaded region}=2\pi-4$

Solve.

$\text{Area of shaded region}=2.28$

In the figure, a square is inscribed in a circle. If the perimeter of the square is , then what is the area of the shaded region?

Explanation

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, let's find the area of the square.

From the given information, we can find the length of a side of the square.

$\text{Perimeter}=4(\text{side})$

$\text{side}=\frac{\text{Perimeter}}{4}$

Substitute in the value of the perimeter to find the length of a side of the square.

$\text{side}=\frac{8}{4}$

Simplify.

$\text{side}=2$

Now recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Substitute in the value of the side of the square to find the area.

$\text{Area of square}=2^2$

Simplify.

$\text{Area of square}=4$

Now, use the Pythagorean theorem to find the length of the diagonal of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}=\sqrt{2(\text{side})^2}$

Simplify.

$\text{Diagonal}=\text{side}\sqrt2$

Substitute in the value of the side of the square to find the length of the diagonal.

$\text{Diagonal}=2\sqrt2$

Recall that the diagonal of the square is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=2\sqrt2$

From the diameter, we can then find the radius of the circle:

$\text{Radius}=\frac{\text{Diameter}}{2}$

$\text{Radius}=\frac{2\sqrt2}{2}$

Simplify.

$\text{Radius}=\sqrt2$

Now, use the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of circle}=\pi\times (\sqrt2)^2$

Simplify.

$\text{Area of circle}=2\pi$

To find the area of the shaded region, subtract the area of the square from the area of the circle.

$\text{Area of Shaded Region}=\text{Area of circle}-\text{Area of square}$

$\text{Area of shaded region}=2\pi-4$

Solve.

$\text{Area of shaded region}=2.28$

A rectangular postage stamp has a width of 3 cm and a height of 12 cm. Find the area of the stamp.

Explanation

A rectangular postage stamp has a width of 3 cm and a height of 12 cm. Find the area of the stamp.

To find the area of a rectangle, we must perform the following:

$A_{Rectangle}=l*w$

Where l and w are our length and width.

This means we need to multiply the given measurements. Be sure to use the right units!

And we have our answer. It must be centimeters squared, because we are dealing with area.

Find the area of a square if it has a diagonal of $\sqrt5$ .

$\frac{5}{2}$

$\frac{15}{2}$

$5\sqrt2$

Explanation

The diagonal of a square is also the hypotenuse of a triangle.

Recall how to find the area of a square:

$\text{Area}=\text{side}^2$

Now, use the Pythagorean theorem to find the area of the square.

$\text{side}^2+\text{side}^2=\text{Diagonal}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{Area}=\text{side}^2=\frac{\text{Diagonal}^2}{2}$

Substitute in the length of the diagonal to find the area of the square.

$\text{Area}=\frac{(\sqrt5)^2}{2}$

Simplify.

$\text{Area}=\frac{5}{2}$

The perimeter of a square is 48. What is the length of its diagonal?

$12\sqrt{2}$

$\sqrt{12}$

$24\sqrt{2}$

$48\sqrt{2}$

$2\sqrt{12}$

Explanation

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

$\sqrt{144+144}=\sqrt{hypotenuse^2}$

$hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}$

A rhombus has a side length of $\frac{5}{6}$ foot, what is the length of the perimeter (in inches).

inches

feet

inches

Explanation

To find the perimeter, first convert $\frac{5}{6}$ foot into the equivalent amount of inches. Since, $\frac{5}{6}=\frac{10}{12}$ and $\frac{12}{12}=1foot$ , $\frac{5}{6}$ is equal to inches.

Then apply the formula , where is equal to the length of one side of the rhombus.

Since,

The solution is:

An isosceles trapezoid has base measurements of and . The perimeter of the trapezoid is . Find the length for one of the two remaining sides.

Explanation

To solve this problem, first note that an isosceles trapezoid has two parallel bases that are nonequivalent in length. Additionally, an isosceles trapezoid must have two nonparallel sides that have equivalent lengths. Since this problem provides the length for both of the bases as well as the total perimeter, the missing sides can be found using the following formula: Perimeter= Base one Base two (leg), where the length of "leg" is one of the two equivalent nonparallel sides.