For the given parallelogram, we will find its perimeter and its area one at a time.

Perimeter

The perimeter of a parallelogram is calculated by adding its four side lengths.

We are given that one side length of the parallelogram is $40$ inches, but we are missing the other three measurements. However, the opposite sides of parallelograms are congruent, so we know that the length of the opposite side is also $40$ inches.

Now, note that one of the sides whose length is missing is the hypotenuse of the right triangle formed on the left of the diagram.

For this right triangle, the lengths of the legs are

36

inches and

27

inches. Let's substitute these values into the Pythagorean Theorem and solve for the hypotenuse

c .

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

SubstituteII

$a = 36$ , $b = 27$

36^{2} + 27^{2} = c^{2}

▼

Solve for

c

CalcPow

Calculate power

1296 + 729 = c^{2}

AddTerms

Add terms

2025 = c^{2}

SqrtEqn

$LHS = RHS$

1154 = c

CalcRoot

Calculate root

45 = c

RearrangeEqn

Rearrange equation

c = 45

Note that we only kept the principal root when solving the equation because

c

is the hypotenuse of a right triangle and it must be non-negative. The length of the hypotenuse is

45

inches. This is also the length of a side of the parallelogram. Therefore, the length of its opposite side is also

45

inches.

Now we can add the four side lengths to obtain the perimeter.

Perimeter : 40 + 45 + 40 + 45 = 170 in .

Area

The area of a parallelogram is the product of its base and its height. In the given parallelogram, we can consider the side whose length is $40$ inches at the base and its corresponding height is $36$ inches.

We can substitute these two values into the formula for the area of a parallelogram and simplify.

A = b h

SubstituteII

$b = 40$ , $h = 36$

A = 40 (36)

Multiply

A = 1440

The area of the parallelogram is

1440

square inches.

Exercise