Area of a Rectangle: Using additional geometric shapes

The rectangle ABCD is shown below.

$BD=25,BC=7$

Calculate the area of the rectangle.

We will use the Pythagorean theorem in order to find the side DC:

$(BC)^2+(DC)^2=(DB)^2$

We begin by inserting the existing data into the theorem:

$7^2+(DC)^2=25^2$

$49+DC^2=625$

$DC^2=625-49=576$

Finally we extract the root:

$DC=\sqrt{576}=24$

168

The trapezoid ABCD and the rectangle ABGE are shown in the figure below.

Given in cm:

AB = 5

BC = 5

GC = 3

Calculate the area of the rectangle ABGE.

Let's calculate side BG using the Pythagorean theorem:

$BG^2+GC^2=BC^2$

We'll substitute the known data:

$BG^2+3^2=5^2$

$BG^2+9=25$

$BG^2=16$

$BG=\sqrt{16}=4$

Now we can calculate the area of rectangle ABGE since we have the length and width:

$5\times4=20$

20

In a rectangular shopping mall they want to place a deltoid-shaped stage.

The length of the rectangle is 30 meters and the width 20 meters.

What is the area of the orange scenario?

We can calculate the area of rectangle ABCD:

$20\times30=600$

Let's divide the deltoid along its length and width and add the following points:

Now we can calculate the area of deltoid PMNK:

$PMNK=\frac{PN\times MK}{2}=\frac{20\times30}{2}=\frac{600}{2}=300$

300 m

Given: the length of a rectangle is 3 greater than its width.

The area of the rectangle is equal to 27 cm².

Calculate the length of the rectangle

The area of the rectangle is equal to length multiplied by width.

Let's set up the data in the formula:

$27=3x\times x$

$27=3x^2$

$\frac{27}{3}=\frac{3x^2}{3}$

$9=x^2$

$x=\sqrt{9}=3$

$x=3$

The area of the rectangle below is equal to 22x.

Calculate x.

The area of the rectangle is equal to the length multiplied by the width.

Let's list the known data:

$22x=\frac{1}{2}x\times(x+8)$

$22x=\frac{1}{2}x^2+\frac{1}{2}x8$

$22x=\frac{1}{2}x^2+4x$

$0=\frac{1}{2}x^2+4x-22x$

$0=\frac{1}{2}x^2-18x$

$0=\frac{1}{2}x(x-36)$

For the equation to be equal, x needs to be equal to 36

$x=36$

ABCD is a parallelogram and AEFD is a rectangle.

AE = 7

The area of AEFD is 35 cm².

CF = 2

What is the area of the parallelogram?

Let's first calculate the sides of the rectangle:

$AEDF=AE\times ED$

Let's input the known data:

$35=7\times ED$

Let's divide the two legs by 7:

$ED=5$

Since AEDF is a rectangle, we can claim that:

ED=FD=7

Let's calculate side CD:

$2+7=9$

Let's calculate the area of parallelogram ABCD:

$ABCD=CD\times ED$

Let's input the known data:

$ABCD=9\times5=45$

45 cm².

Shown below is the rectangle ABCD.

Given in cm:

AK = 5

DK = 4

The area of the rectangle is 24 cm².

Calculate the side AB.

Let's look at triangle ADK to calculate side AD:

$AD^2+DK^2=AK^2$

Let's input the given data:

$AD^2+4^2=5^2$

$AD^2+16=25$

We'll move 16 to the other side and change the appropriate sign:

$AD^2=25-16$

$AD^2=9$

We'll take the square root and get:

$AD=3$

Since AD is a side of rectangle ABCD, we can calculate side AB as follows:

$S=AB\times AD$

Let's input the given data:

$24=3\times AB$

We'll divide both sides by 3:

$AB=8$

8

Given that: the area of the rectangle is equal to 36.

$AE=\frac{1}{4}AB$

Find the size of AE.

The area of rectangle ABCD equals length multiplied by width.

Let's input the known data into the formula in order to find side AB:

$36=3\times AB$

Let's divide both sides by 3:

$AB=12$

Since we are given that AE equals a quarter of AB, we can substitute the known data and calculate side AE:

$AE=\frac{1}{4}AB=\frac{1}{4}\times12=3$

3

The area of the rectangle is equal to 70.

$EB=\frac{1}{5}AB$

Calculate the length of EB.

The area of rectangle ABCD equals length multiplied by width.

Let's use the known data in the formula to find side AB:

$70=7\times AB$

Let's divide both sides by 7:

$AB=10$

Since we are given that EB equals one-fifth of AB, we can use the known data and calculate side EB:

$EB=\frac{1}{5}AB=\frac{1}{5}\times10=2$

2

Given the rectangle ABCD

Given BC=X and the side AB is larger by 4 cm than the side BC.

The area of the triangle ABC is 8X cm².

What is the area of the rectangle?

Let's calculate the area of triangle ABC:

$8x=\frac{(x+4)x}{2}$

Multiply by 2:

$16x=(x+4)x$

Divide by x:

$16=x+4$

Let's move 4 to the left side and change the sign accordingly:

$16-4=x$

$12=x$

Now let's calculate the area of the rectangle, multiply the length and width where BC equals 12 and AB equals 16:

$16\times12=192$

192

Calculate the area of the rectangle below in terms of a and b.

Let us begin by reminding ourselves of the formula to calculate the area of a rectangle: width X length

$S=w⋅h$

When:

S = area

w = width

h = height

We take data from the sides of the rectangle in the figure.$w=b+8$$h=a+3$

We then substitute the above data into the formula in order to calculate the area of the rectangle:

$S=w⋅h = (b+8)(a+3)$

We use the formula of the extended distributive property:

$(a+b)(c+d)=ac+ad+bc+bd$

We substitute once more and solve the problem as follows:

$S=(b+8)(a+3)=(b)(a)+(b)(3)+(8)(a)+(8)(3)$

$(b)(a)+(b)(3)+(8)(a)+(8)(3)=ab+3b+8a+24$

Therefore, the correct answer is option B: ab+8a+3b+24.

Keep in mind that, since there are only addition operations, the order of the terms in the expression can be changed and, therefore,

$ab+3b+8a+24=ab+8a+3b+24$

ab + 8a + 3b + 24

Express the area of the rectangle below in terms of y and z.

Let us begin by reminding ourselves of the formula to calculate the area of a rectangle: width X height

$S=w⋅h$

Where:

S = area

w = width

h = height

We must first extract the data from the sides of the rectangle shown in the figure.

$w=3y$$h=y+3z$

We then insert the known data into the formula in order to calculate the area of the rectangle:

$S=w⋅h=(y+3z)(3y)$

We use the distributive property formula:

$a\left(b+c\right)=ab+ac$

We substitute all known data and solve as follows:

$S=(y+3z)(3y)=(3y)(y+3z)$

$(3y)(y+3z)=(3y)(y)+(3y)(3z)$

$(3y)(y)+(3y)(3z)=3y^2+9yz$

Keep in mind that because there is a multiplication operation, the order of the terms in the expression can be changed, hence:

$(y+3z)(3y)=(3y)(y+3z)$

Therefore, the correct answer is option D: $3y^2+9yz$

$3y^2+9yz$

The area of the rectangle in the drawing is 28X cm².

What is the area of the circle?

Let's draw the center of the circle and we can divide the diameter of the circle into two equal radii

Now let's calculate the length of the radii as follows:

$7\times2r=28x$

$14r=28x$

We'll divide both sides by 14:

$r=\frac{28}{14}x$

$r=2x$

Let's calculate the circumference of the circle:

$P=2\pi\times r=2\pi\times2x=4\pi x$

$4\pi x$

Calculate the area of the rectangle

Let's begin by reminding ourselves of the formula to calculate the area of a rectangle: width X length

$S=w⋅h$

Where:

S = area

w = width

h = height

We extract the data from the sides of the rectangle in the figure.

$w=x+5$$h=y+2$

We then substitute the above data into the formula in order to calculate the area of the rectangle:

$S=w⋅h=(x+5)(y+2)$

We use the formula of the extended distributive property:

$(a+b)(c+d)=ac+ad+bc+bd$

We once again substitute and solve the problem as follows:

$S=(x+5)(y+2)=(x)(y)+(x)(2)+(5)(y)+(5)(2)$

$(x)(y)+(x)(2)+(5)(y)+(5)(2)=xy+2x+5y+10$

Therefore, the correct answer is option C: xy+2x+5y+10.

^{$xy+2x+5y+10$}

Below is a hexagon that contains a rectangle inside it.
The area of the rectangle is 28 cm².

What is the area of the hexagon?

Since we are given the area of the rectangle, let's find the length of the missing side:

$7\times a=28$

We'll divide both sides by 7 and get:

$a=4$

Since in a hexagon all sides are equal to each other, each side is equal to 4.

Now let's calculate the area of the hexagon:

$\frac{6\times a^2\times\sqrt{3}}{4}$

$\frac{6\times4^2\times\sqrt{3}}{4}$

Let's simplify the exponent in the denominator of the fraction and we'll get:

$6\times4\times\sqrt{3}=24\times\sqrt{3}=41.56$

41.56

Given the rectangle ABCD

AB=X

The ratio between AB and BC is $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal A the rectangle in m

Check the correct argument:

Given that:

$\frac{AB}{BC}=\sqrt{\frac{x}{2}}$

Given that AB equals X

We will substitute accordingly in the formula:

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$x\sqrt{2}=BC\sqrt{x}$

$\frac{x\sqrt{2}}{\sqrt{x}}=BC$

$\frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC$

$\sqrt{x}\times\sqrt{2}=BC$

Now let's focus on triangle ABC and use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

Let's substitute the known values:

$x^2+(\sqrt{x}\times\sqrt{2})^2=m^2$

$x^2+x\times2=m^2$

We'll add 1 to both sides:

$x^2+2x+1=m^2+1$

$(x+1)^2=m^2+1$

$m^2+1=(x+1)^2$

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal to$\sqrt{\frac{x}{2}}$

We mark the length of the diagonal $A$ with $m$

Check the correct argument:

Let's find side BC

Based on what we're given:

$\frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}$

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$\sqrt{2}x=\sqrt{x}BC$

Let's divide by square root x:

$\frac{\sqrt{2}\times x}{\sqrt{x}}=BC$

$\frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC$

Let's reduce the numerator and denominator by square root x:

$\sqrt{2}\sqrt{x}=BC$

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

$AB^2+BC^2=AC^2$

Let's substitute what we're given:

$x^2+(\sqrt{2}\sqrt{x})^2=m^2$

$x^2+2x=m^2$

$x^2+2x=m^2$

The area of the rectangle below is equal to 45.

$ED=\frac{1}{3}AB$

Calculate the size of ED.

3

The perimeter of a rectangle is 14 cm.

The area of the rectangle is 12 cm².

What are the lengths of its sides?

3, 4

The height of the house in the drawing is $12x+9$

its width $x+2y$

Given the ceiling height is half the height of the square section.

Express the area of the house shape in the drawing band x and and.

$3x^2+8xy+1\frac{1}{2}x+4y^2+3y$