How to find the length of the second side of a rectangle. How to find the sides of a rectangle if the area and perimeter are known

Instructions

For example, you know that the length of one of the sides (a) is 7 cm, and perimeter rectangle(P) is equal to 20 cm. Since perimeter of any figure is equal to the sum of the lengths of its sides, and rectangle opposite sides are equal, then its perimeter a will look like this: P = 2 x (a + b), or P = 2a + 2b. From this formula it follows that you can find the length of the second side (b) using a simple operation: b = (P – 2a) : 2. So, in our case, side b will be equal to (20 – 2 x 7) : 2 = 3 cm .

Now, knowing the lengths of both adjacent sides (a and b), you can substitute them into the area formula S = ab. In this case rectangle will be equal to 7x3 = 21. Please note that the units of measurement will no longer be , but square centimeters, since you also multiplied the lengths of the two sides of their units of measurement (centimeters) by each other.

Sources:

What is the perimeter of a rectangle?

A flat figure consisting of four sides and four right angles. Of all the figures square rectangle have to be calculated more often than others. This and square apartments, and square garden plot, and square table or shelf surfaces. For example, to simply wallpaper a room, they calculate square its rectangular walls.

Instructions

By the way, from rectangle can be easily calculated square. It is enough to complete the rectangular one to rectangle so that the hypotenuse becomes a diagonal rectangle. Then it will be obvious that square such rectangle is equal to the product of the legs of the triangle, and square of the triangle itself, accordingly, is equal to half the product of the legs.

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A special case of a parallelogram - a rectangle - is known only in Euclidean geometry. U rectangle All angles are equal, and each of them separately makes 90 degrees. Based on private properties rectangle, and also from the properties of a parallelogram about the parallelism of opposite sides can be found sides figures along given diagonals and the angle from their intersection. Calculating sides rectangle is based on additional constructions and application of the properties of the resulting figures.

Instructions

Use the letter A to mark the point of intersection of the diagonals. Consider the EFA formed by the constructs. According to property rectangle its diagonals are equal and bisected by the intersection point A. Calculate the values of FA and EA. Since triangle EFA is isosceles and its sides EA and FA are equal to each other and respectively equal to half of the diagonal EG.

Next, calculate the first EF rectangle. This side is the third unknown side of the triangle EFA under consideration. According to the cosine theorem, use the appropriate formula to find the side EF. To do this, substitute the previously obtained values of the sides FA EA and the cosine of the known angle between them α into the cosine formula. Calculate and record the resulting EF value.

Find the other side rectangle F.G. To do this, consider another triangle EFG. It is rectangular, where the hypotenuse EG and leg EF are known. According to the Pythagorean theorem, find the second leg of FG using the appropriate formula.

Refers to the simplest flat geometric figures and is one of the special cases of a parallelogram. A distinctive feature of such a parallelogram is right angles at all four vertices. Limited by parties rectangle square can be calculated in several ways, using the dimensions of its sides, diagonals and angles between them, the radius of the inscribed circle, etc.

Instructions

If the magnitude of the angle (α) that makes up the diagonal is known rectangle on one of its sides, as well as the length (C) of this diagonal, then to calculate the area you can use the definitions of trigonometric in a rectangular. The right triangle here is formed by two sides of the quadrilateral and its diagonal. From the definition of cosine it follows that the length of one of the sides will be equal to the product of the length of the diagonal and the angle, the value is known. From the definition of sine, we can derive the formula for the length of the other side - it is equal to the product of the length of the diagonal and the sine of the same angle. Substitute these identities into the formula from the previous step, and it turns out that to find the area you need to multiply the sine and cosine of a known angle, as well as the length of the diagonal rectangle: S=sin(α)*cos(α)*С².

If, in addition to the diagonal length (C) rectangle If the magnitude of the angle (β) formed by the diagonals is known, then to calculate the area of the figure you can also use one of the trigonometric functions - sine. Square the length of the diagonal and multiply the result by half the sine of the known angle: S=С²*sin(β)/2.

If the (r) of the circle inscribed in the rectangle is known, then to calculate the area, raise this value to the second power and quadruple the result: S=4*r². A quadrilateral into which it is possible will be a square, and the length of its side is equal to the diameter of the inscribed circle, that is, twice the radius. The formula is obtained by substituting the lengths of the sides, expressed in terms of the radius, into the identity from the first step.

If the lengths (P) and one of the sides (A) are known rectangle, then to find the area inside this perimeter, calculate half the product of the side length and the difference between the length of the perimeter and the two lengths of this side: S=A*(P-2*A)/2.

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Not only students in geometry lessons are faced with the task of finding the perimeter or area of a polygon. Sometimes it happens to be solved by an adult. Have you ever had to calculate the required amount of wallpaper for a room? Or maybe you measured the length of your summer cottage in order to enclose it with a fence? Thus, knowledge of the basics of geometry is sometimes indispensable for the implementation of important projects.

4a, where a is the side of a square or rhombus. Then the length sides equal to one fourth of the perimeter: a = p/4.

This problem can also be easily solved for a triangle. He has three of the same length sides, so the perimeter p of an equilateral triangle is 3a. Then the side of the equilateral triangle is a = p/3.

For the remaining figures you will need additional data. For example, you can find sides, knowing its perimeter and area. Suppose the length of the two opposite sides of the rectangle is a, and the length of the other two sides is b. Then the perimeter p of the rectangle is 2(a+b), and the area s is equal to ab. We get a system with two unknowns:
p = 2(a+b)
s = ab. Express from the first equation a: a = p/2 - b. Substitute into the second and find b: s = pb/2 - b². The discriminant of this equation is D = p²/4 - 4s. Then b = (p/2±D^1/2)/2. Discard the root that is less than zero and substitute in for sides a.

Sources:

Find the sides of a rectangle

If you know the value of a, then you can say that you have solved the quadratic equation, because its roots will be found very easily.

You will need

-discriminant formula for a quadratic equation;
-knowledge of multiplication tables

Instructions

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Helpful advice

The discriminant of a quadratic equation can be positive, negative, or equal to 0.

Sources:

Solving Quadratic Equations
discriminant even

Instructions

Tip 4: How to find the perimeter of an equilateral triangle

An equilateral triangle, along with a square, is perhaps the simplest and most symmetrical figure in planimetry. Of course, all relations that are valid for an ordinary triangle are also true for an equilateral triangle. However, for a regular triangle, all formulas become much simpler.

You will need

calculator, ruler

Instructions

To measure the length of one of its sides and multiply the measurement by three. This can be written as follows:

Prt = Ds * 3,

Prt – perimeter of the triangle,
Ds is the length of any of its sides.

The perimeter of the triangle will be in the same dimensions as the length of its side.

Since an equilateral triangle has a high degree of symmetry, one of the parameters is sufficient to calculate its perimeter. For example, area, height, inscribed or circumscribed circle.

If you know the radius of the incircle of an equilateral triangle, then use the following formula to calculate its perimeter:

Prt = 6 * √3 * r,

where: r is the radius of the inscribed circle.
This rule follows from the fact that the radius of the incircle of an equilateral triangle is expressed in terms of the length of its side by the following relation:
r = √3/6 * Ds.

To calculate the perimeter in terms of the circumradius, use the formula:

Prt = 3 * √3 * R,

where: R is the radius of the circumscribed circle.
This is easily derived from the fact that the circumradius of a regular triangle is expressed through the length of its side by the following relation: R = √3/3 * Ds.

To calculate the perimeter of an equilateral triangle through a known area, use the following relationship:
Srt = Dst² * √3 / 4,
where: Sрт – area of an equilateral triangle.
From here we can deduce: Dst² = 4 * Sрт / √3, therefore: Dst = 2 * √(Sрт / √3).
Substituting this ratio into the perimeter formula through the length of the side of an equilateral triangle, we obtain:

Prt = 3 * Dst = 3 * 2 * √(Srt / √3) = 6 * √Sst / √(√3) = 6√Sst / 3^¼.

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A square is a geometric figure consisting of four sides of equal length and four right angles, each of which is 90°. Determination of area or perimeter a quadrilateral, any kind at that, is required not only when solving geometry problems, but also in everyday life. These skills can become useful, for example, during repairs when calculating the required amount of materials - coverings for floors, walls or ceilings, as well as for laying out lawns and beds, etc.

So, first, let's look at the formulas for finding area and perimeter:

1) S = a * b = 56 cm2;

2) P = 2a + 2b = 30 cm.

After all, we know that a rectangle has two identical sides.

Thus, we need to solve a system of two equations:

From this we see that one side is 7 and the other is 8.

Knowing the formulas for the perimeter of a rectangle and its area, the sides are sought in the form of solving a system of two equations. First, we express the value of one side through the other and, for example, the area. It looks like this: A = S / B = 56 / B

Then we substitute this expression for the letter A in the equation for the perimeter:

P=2(56/V + V)=30

We get that 56/B+B=15

In this equation, you don’t even need to solve it - anyone familiar with the multiplication table can immediately see that 56 is the product of 7 and 8, and since the sum of these numbers is just 15, then they are the values of the sides of the rectangle we need.

You can try to solve this problem by creating a system of equations.

The perimeter of the rectangle is: p=2a+2b;

The area of the rectangle is: s=a*b;

Since we know the perimeter and area, we immediately substitute the numbers:

Express b in terms of a in the second equation:

And substitute 56/a instead of b in the first equation:

Multiply both sides by a:

We get a quadratic equation:

Finding the roots of this quadratic equation:

(15(15-4*1*56))/2*1 = (15(225-224))/2 = (151)/2 = (151)/2

It turns out that the roots of this equation are:

a1=(15+1)/2=16/2=8;

a2=(15-1)/2=14/2=7;

It turns out that we have 2 possible options for rectangles.

Let's remember what we expressed: b=56/a;

From here we find possible b:

b1=56/a1=56/8=7;

b2=56/a2=56/7=8;

As it turned out, these two different rectangles are one and the same; you can simply achieve a perimeter of 30 with an area of 56:

If a=7 and b=8.

Or vice versa: a=8 and b=7.

That is, in essence, we have the same rectangle, just in one version the vertical side is larger than the horizontal, and in the other, on the contrary, the horizontal is larger than the vertical.

Answer: one side is 7 centimeters, and the other is 8 centimeters.

Let's remember school geometry:
The perimeter of a rectangle is the sum of the lengths of all sides, and the area of a rectangle is the product of its two adjacent sides (length by width).
In this case, we know both the Area and Perimeter of the rectangle. They are 56 cm^2 and 30 cm, respectively.
So, the solution:
S - area = a x b;
P - perimeter = a + b + a + b = 2a + 2b;
30 = 2 (a + b);
Let's make a substitution:
56 = (15 - b) x b;
56 = 15 b - b^2;
b^2 - 15b + 56 = 0.
We got a quadratic equation, solving which we get: b1 = 8, b2 = 7.
We find the other side of the rectangle:
a1 = 15 - 8 = 7;
a2 = 15 - 7 = 8.
Answer: The sides of the rectangle are 8 and 7 cm or 7 and 8 cm.
If the perimeter of a rectangle is P = 30 cm and its area is S = 56 cm, then its sides will be equal:

a - one side, b - the other side of the rectangle.
Having solved this system, we come to the conclusion that side a will be equal to 7 cm, and side b will be equal to 8 cm.
a = 7 cm b = 8 cm.
Given: S = 56 cm
P = 30 cm
Sides=?
Solution:
Let the sides of the rectangle be a and b.
Then: area S = a * b, perimeter P=2*(a + b),
We get a system of equations:
(a*b=56 ? (ab=56
(2(a+b)=30, (a+b=15, expressing b through a we get a quadratic equation:
b=15-a, a^2 -15a +56 =0 , solving which, we get:
b1=8, b2=7. That is, the sides of the rectangle: a=7,b=8, or vice versa: a=8,b=7.

To solve the problem, you need to create a system of equations and solve it

we get a quadratic equation that can be easily solved if we substitute the values of perimeter and area into it

The discriminant is 1 and the equation has two roots 7 and 8, therefore one of the sides equal to 7 cm, the other 8 cm or vice versa.

I specifically wrote out the discriminant here, since it is very easy to navigate

if in the condition of the problem of finding the sides of a rectangle, the value of the perimeter and area are specified so that this discriminant more than zero, then we have rectangle;

if discriminant equal to zero- then we have square(P=30, S=56.25, square with side 7.5);

if discriminant less than zero, then like this rectangle does not exist(P=20, S=56 - no solution)

Perimeter 30, area 56. Let's call the sides of the rectangle a and c. Then we can create the following equations:

Let's denote one side by the letter X, the other by the letter Y.

The area of a rectangle is calculated by multiplying the lengths of the sides, so we can formulate the first equation:

The perimeter is the sum of the lengths of the sides, therefore the second equation is:

We obtain a system of two equations.

Using the first equation, select X: X=56:Y, substitute this into the second equation:

2*56:Y+2Y=30 From here it’s easy to find the value of Y: Y=7, then X=8.

I found another solution:

It is known that the perimeter of a rectangle is 30 and the area is 56, then:

perimeter = 2*(length + width) or 2L + 2W

area= length * width or L * W

2L + 2W = 30 (divide both parts by 2)

L * (15 - L) = 56

To be honest, I didn’t quite understand the solution, but I think anyone who hasn’t completely forgotten mathematics will figure it out.

Side A=7, side B=8

Instructions

Length rectangle can be found in several ways. It all depends on the source data.

Option one is perhaps the simplest.

If the width is known rectangle and its area, we use the area formula. It is known that the area rectangle product of width and length rectangle.

Perimeter rectangle it is possible to find by adding the width and length values and multiplying the resulting number by two. We find the unknown side.

We divide the perimeter by two and subtract the width from the resulting figure.

If only the width is known rectangle and the length of the diagonal, you can use the Pythagorean theorem. Divide the rectangle into two equal rectangles.

The next method: the angle between the diagonals is known rectangle and diagonal. Consider the triangle formed rectangle and halves of diagonals. Using the cosine theorem you will find this side rectangle.

Sources:

find the width of the rectangle
What is the length of a rectangle if its width is known?

Each of us learned about what a perimeter is in elementary school. Finding the sides of a square with a known perimeter usually does not cause problems even for those who graduated from school a long time ago and managed to forget the mathematics course. However, not everyone can solve a similar problem regarding a rectangle or right triangle without prompting.

Instructions

Suppose that there is a right triangle with sides a, b and c, in which one of the angles is 30 and the other is 60. The figure shows that a = c*sin?, and b = c*cos?. Knowing that the perimeter of any figure, in and triangle, is equal to the sum of all its sides, we obtain:a+b+c=c*sin ?+c*cos+c=pFrom this expression we can find the unknown side c, which is the hypotenuse for the triangle . So what's the angle? = 30, after transformation we get: c*sin ?+c*cos ?+c=c/2+c*sqrt(3)/2+c=p It follows that c=2p/Accordingly, a = c*sin ?= p/,b=c*cos ?=p*sqrt(3)/

As mentioned above, the diagonal of a rectangle divides it into two right triangles with angles of 30 and 60 degrees. Since it is equal to p=2(a + b), width a and length b of a rectangle can be found based on the fact that the diagonal is the hypotenuse of right triangles:a = p-2b/2=p/2
b= p-2a/2=p/2These two equations are rectangles. From them, the length and width of this rectangle are calculated, taking into account the resulting angles when drawing its diagonal.

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note

How to find the length of a rectangle if the perimeter and width are known? Subtract twice the width from the perimeter, then we get twice the length. Then we divide it in half to find the length.

Helpful advice

Even from elementary school, many people remember how to find the perimeter of any geometric figure: just find out the length of all its sides and find their sum. It is known that in a figure such as a rectangle, the lengths of the sides are equal in pairs. If the width and height of a rectangle are the same length, then it is called a square. Typically, the length of a rectangle is the largest side, and the width is the smallest.

Sources:

what is the perimeter width in 2019

Tip 3: How to find the area of a triangle and a rectangle

Triangle and rectangle are the two simplest plane geometric figures in Euclidean geometry. Inside the perimeters formed by the sides of these polygons, there is a certain section of the plane, the area of which can be determined in many ways. The choice of method in each specific case will depend on the known parameters of the figures.

Instructions

Use one of the formulas using trigonometric formulas to find the area of a triangle if the values of one or more angles in are known. For example, with a known angle (α) and the lengths of the sides that make it up (B and C), the area (S) can be calculated using the formula S=B*C*sin(α)/2. And with the values of all angles (α, β and γ) and the length of one side in addition (A), you can use the formula S=A²*sin(β)*sin(γ)/(2*sin(α)). If, in addition to all angles, (R) of the circumscribed circle is known, then use the formula S=2*R²*sin(α)*sin(β)*sin(γ).

If the angles are not known, then you can use trigonometric functions to find the area of the triangle. For example, if (H) is drawn from a side that also knows (A), then use the formula S=A*H/2. And if the lengths of each side (A, B and C) are given, then first find the semi-perimeter p=(A+B+C)/2, and then calculate the area of the triangle using the formula S=√(p*(p-A)* (p-B)*(p-C)). If, in addition to (A, B and C), the radius (R) of the circumscribed circle is known, then use the formula S=A*B*C/(4*R).

To find the area of a rectangle, you can also use trigonometric functions - for example, if you know the length of its diagonal (C) and the size of the angle it makes on one of the sides (α). In this case, use the formula S=С²*sin(α)*cos(α). And if the lengths of the diagonals (C) and the size of the angle they make (α) are known, then use the formula S=C²*sin(α)/2.

You can do without trigonometric functions when finding the area of a rectangle if you know the lengths of its perpendicular sides (A and B) - you can use the formula S=A*B. And if the length of the perimeter (P) and one side (A) is given, then use the formula S=A*(P-2*A)/2.

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Division is one of the basic arithmetic operations. It is the opposite of multiplication. As a result of this action, you can find out how many times one of the given numbers is contained in another. In this case, division can replace an infinite number of subtractions of the same number. Problem books regularly contain the task of finding an unknown dividend.

You will need

- calculator;
- a sheet of paper and a pencil.

Instructions

Label the unknown dividend as x. Write known data either using given numbers or alphabetic symbols. For example, a task might look like this: x:a=b. Moreover, a and b can be any numbers, both , and . A quotient in the form of an integer means that the division is performed without a remainder. To find the dividend, multiply the quotient by the divisor. The formula will look like this: x=a*b.

If the divisor or quotient is not an integer, remember the features of multiplying fractions and decimals. In the first case, the numerators and denominators are multiplied. If one number is an integer and the other is a simple fraction, the numerator of the second is multiplied by the first. Decimals are multiplied in the same way as whole numbers, but the number of digits to the right of the decimal point are added together, and the trailing zero is included.

Let us assume that two sides of a rectangle that have one common point (i.e. its length) are specified by the coordinates of three points A(X₁,Y₁), B(X₂,Y₂) and C(X₃,Y₃). The fourth point need not be considered - its coordinates do not affect in any way. The length of the projection of side AB onto the abscissa axis will be equal to the difference between the corresponding coordinates of these points (X₂-X₁). The length of the projection onto the ordinate axis is determined similarly: Y₂-Y₁. This means that the length of the side itself, according to the Pythagorean theorem, can be found as the square root

Definition.

Rectangle is a quadrilateral in which two opposite sides are equal and all four angles are equal.

The rectangles differ from each other only in the ratio of the long side to the short side, but all four corners are right, that is, 90 degrees.

The long side of a rectangle is called rectangle length, and the short one - rectangle width.

The sides of a rectangle are also its heights.

Basic properties of a rectangle

A rectangle can be a parallelogram, a square or a rhombus.

1. The opposite sides of the rectangle have the same length, that is, they are equal:

AB = CD, BC = AD

2. Opposite sides of the rectangle are parallel:

3. The adjacent sides of a rectangle are always perpendicular:

AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB

4. All four corners of the rectangle are straight:

∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°

5. The sum of the angles of a rectangle is 360 degrees:

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

6. The diagonals of a rectangle have the same length:

7. The sum of the squares of the diagonal of a rectangle is equal to the sum of the squares of the sides:

2d 2 = 2a 2 + 2b 2

8. Each diagonal of a rectangle divides the rectangle into two identical figures, namely right triangles.

9. The diagonals of the rectangle intersect and are divided in half at the intersection point:

		AO=BO=CO=DO=	d
			2

10. The point of intersection of the diagonals is called the center of the rectangle and is also the center of the circumcircle

11. The diagonal of a rectangle is the diameter of the circumcircle

12. You can always describe a circle around a rectangle, since the sum of opposite angles is 180 degrees:

∠ABC = ∠CDA = 180° ∠BCD = ∠DAB = 180°

13. A circle cannot be inscribed in a rectangle whose length is not equal to its width, since the sums of the opposite sides are not equal to each other (a circle can only be inscribed in a special case of a rectangle - a square).

Sides of a rectangle

Definition.

Rectangle length is the length of the longer pair of its sides. Rectangle width is the length of the shorter pair of its sides.

Formulas for determining the lengths of the sides of a rectangle

1. Formula for the side of a rectangle (length and width of the rectangle) through the diagonal and the other side:

a = √ d 2 - b 2

b = √ d 2 - a 2

2. Formula for the side of a rectangle (length and width of the rectangle) through the area and the other side:

b = dcos	β
	2

Diagonal of a rectangle

Definition.

Diagonal rectangle Any segment connecting two vertices of opposite corners of a rectangle is called.

Formulas for determining the length of the diagonal of a rectangle

1. Formula for the diagonal of a rectangle using two sides of the rectangle (via the Pythagorean theorem):

d = √ a 2 + b 2

2. Formula for the diagonal of a rectangle using the area and any side:

4. Formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:

d = 2R

5. Formula for the diagonal of a rectangle in terms of the diameter of the circumcircle:

d = D o

6. Formula for the diagonal of a rectangle using the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:

8. Formula for the diagonal of a rectangle through the sine of the acute angle between the diagonals and the area of the rectangle

d = √2S: sin β

Perimeter of a rectangle

Definition.

Perimeter of a rectangle is the sum of the lengths of all sides of a rectangle.

Formulas for determining the length of the perimeter of a rectangle

1. Formula for the perimeter of a rectangle using two sides of the rectangle:

P = 2a + 2b

P = 2(a + b)

2. Formula for the perimeter of a rectangle using area and any side:

P=	2S + 2a 2	=	2S + 2b 2
	a		b

3. Formula for the perimeter of a rectangle using the diagonal and any side:

P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)

4. Formula for the perimeter of a rectangle using the radius of the circumcircle and any side:

P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)

5. Formula for the perimeter of a rectangle using the diameter of the circumscribed circle and any side:

P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)

Area of a rectangle

Definition.

Area of a rectangle called the space limited by the sides of the rectangle, that is, within the perimeter of the rectangle.

Formulas for determining the area of a rectangle

1. Formula for the area of a rectangle using two sides:

S = a b

2. Formula for the area of a rectangle using the perimeter and any side:

5. Formula for the area of a rectangle using the radius of the circumcircle and any side:

S = a √4R 2 - a 2= b √4R 2 - b 2

6. Formula for the area of a rectangle using the diameter of the circumcircle and any side:

S = a √D o 2 - a 2= b √D o 2 - b 2

Circle circumscribed around a rectangle

Definition.

A circle circumscribed around a rectangle is a circle passing through the four vertices of a rectangle, the center of which lies at the intersection of the diagonals of the rectangle.

Formulas for determining the radius of a circle circumscribed around a rectangle

1. Formula for the radius of a circle circumscribed around a rectangle through two sides:

How to find the length of the second side of a rectangle. How to find the sides of a rectangle if the area and perimeter are known

Tip 4: How to find the perimeter of an equilateral triangle

Answer: one side is 7 centimeters, and the other is 8 centimeters.

Given: S = 56 cm

P = 30 cm

Sides=?

Solution:

Tip 3: How to find the area of ​​a triangle and a rectangle

Basic properties of a rectangle

Sides of a rectangle

Formulas for determining the lengths of the sides of a rectangle

Diagonal of a rectangle

Formulas for determining the length of the diagonal of a rectangle

Perimeter of a rectangle

Formulas for determining the length of the perimeter of a rectangle

Area of ​​a rectangle

Formulas for determining the area of ​​a rectangle

Circle circumscribed around a rectangle

Formulas for determining the radius of a circle circumscribed around a rectangle

Tip 3: How to find the area of a triangle and a rectangle

Area of a rectangle

Formulas for determining the area of a rectangle