Math - How to find the area of a rectangle

A rectangle has sides of units and units. If the perimeter of the rectangle is units, what is its area?

units squared

Explanation

Since a rectangle has pairs of equal-length sides, multiplying each side by and adding the products together gives the perimeter of the rectangle. Use this fact to set up an equation with the given information about the rectangle's sides and perimeter. Solving for in this equation will provide necessary information for finding the rectangle's area:

$\rightarrow 4x+2 + 14 = 104$

$\rightarrow 4x+ 16 = 104$

$\rightarrow 4x = 88$

$\rightarrow x = 22$

Multiplying the measure of the long side of the rectangle by the measure of the short side of the rectangle gives the rectangle's area. The length of the long side is given by substituting the solution for into the given expression that defines its length. The short side is , giving the following equation to calculate the area:

$A = (2(22)+1) \times 7$

$\rightarrow A = 45 \times 7$

$\rightarrow A = 315$ units squared

Erin is getting ready to plant her tulip garden. She wants to plant two tulips per square foot of garden. If her rectangular garden is enclosed by 24 feet of fencing, and the length of the fence is twice as long as its width, how many tulips will Erin plant?

Explanation

We know that the following represents the formula for the perimeter of a rectangle:

In this particular case, we are told that the length of the fence is twice as long as the width. We can write this as the following expression:

Use this information to substitute in a variable for the length that matches the variable for width in our perimeter equation.

We also know that the length is two times the width; therefore, we can write the following:

$l = 2\times4$

The area of a rectangle is found by using this formula:

$A=8\times4$

The area of the garden is 32 square feet. Erin will plant two tulips per square foot; thus, she will plant 64 tulips.

If the width of a rectangle is 8 inches, and the length is half the width, what is the area of the rectangle in square inches?

Explanation

the length of the rectangle is half the width, and the width is 8, so the length must be half of 8, which is 4.

The area of the rectangle can be determined from multiplying length by width, so,

4 x 8 = 32 inches squared

George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 $ft^{2}$ and costs $40. One quart of paint covers 100 $ft^{2}$ and costs $15. How much money will he spend on the blue paint?

Explanation

The area of the walls is given by $A = 2wh + 2lh = 2(12)(10) + 2(15)(10) = 540 ft^{2}$

One gallon of paint covers 400 $ft^{2}$ and the remaining 140 $ft^{2}$ would be covered by two quarts.

So one gallon and two quarts of paint would cost

Angela is carpeting a rectangular conference room that measures 20 feet by 30 feet. If carpet comes in rectangular pieces that measures 5 feet by 4 feet, how many carpet pieces will she need to carpet the entire room?

600

Explanation

First, we need to find the area of the room. Because the room is rectangular, we can multiply 20 feet by 30 feet, which is 600 square feet. Next, we need to know how much space one carpet piece covers. Because the carpet pieces are also rectangular, we can multiply 4 feet by 5 feet to get 20 feet. To determine how many pieces of carpet Angela will need, we must divide the total square footage of the room (600 feet) by the square footage covered by one carpet piece (20 feet). 600 divided by 20 is 30, so Angela will need 30 carpet pieces to carpet the entire room.

Rectangle_with_diagonal

A rectangle has a diagonal of and a width of . What is the area of the rectangle?

$\dpi{100} 31.22cm^{2}$

$22.31cm^{2}$

$40.00cm^{2}$

Not enough information to solve

$26.26cm^{2}$

Explanation

We are given the rectangle's diagonal $\dpi{100} \dpi{100} (d=8cm)$ and width $\dpi{100} (w=5cm)$ and are asked to find its area. The diagonal forms two right triangles within the rectangle; therefore, we can use the Pythagorean theorem to find the length of the rectangle's missing side. Then we can use the formula to find the are of the rectangle.

$a^{2}+b^{2}=c^{2}$

In our case, we can rename the variables to match our triangle.

$w^{2}+l^{2}=d^{2}$

$\dpi{100} \dpi{100} \dpi{100} (5cm)^{2}+l^{2}=(8cm)^{2}$

$\dpi{100} l^{2}=64cm^{2}-25cm^{2}$

$\dpi{100} \dpi{100} \dpi{100} \sqrt{l^{2}}=\sqrt{39cm^{2}}$

$\dpi{100} l=\sqrt{39cm^{2}}$

Now we can calculate the area.

$A=(\sqrt{39cm^{2}})*5cm$

$\rightarrow 31.22cm^{2}$

Para-rec1

Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?

272

544

176

352

Explanation

Para-rec2

Para-rec3

Joey is working in his yard. In the middle of his rectangular yard he places a round cement fountain. The fountain has a diameter of . One bag of grass seed covers square feet. How many bags of grass seed will Joey need to cover his yard?

Explanation

First, find the area of the rectangular yard:

square feet

Next, find the area of the round cement fountain:

$\pi r^{2}=\pi (9)^{2}\approx 3.14(9)^{2}\approx 254$ square feet

Then find the difference between the two:

square feet

Now, to get the number of grass seed bags needed, divide the area by to get approximately bags. Because one can't purchase a partial bag, the correct answer is the next largest whole number, or bags of grass seed.