Finding the Dimensions of a Rectangle Given Its Perimeter and Area

When you have given the perimeter and area of a rectangle, finding its dimensions requires a little bit of algebra and solving a quadratic equation. In this article, we will guide you through the process of determining the dimensions of a rectangle with a known perimeter of 40 cm and an area of 84 square cm.

Understanding the Problem

Given:

Perimeter of a rectangle 40 cm Area of the rectangle 84 square cm

Solving for Dimensions Using Perimeter and Area

The formulas for perimeter and area of a rectangle are:

Perimeter (P) 2(l w)

Area (A) l × w

Where:

l length of the rectangle w width of the rectangle

Step 1: Setting Up Equations

First, let's set up the equations based on the given information:

Perimeter equation: 2(l w) 40

Area equation: l × w 84

Step 2: Simplifying the Equations

Solving the perimeter equation for w:

2(l w) 40

l w 20

w 20 - l

Now, substitute w 20 - l into the area equation:

l × (20 - l) 84

Step 3: Forming and Solving the Quadratic Equation

Rearrange the equation to form a quadratic equation:

20l - l^2 84

-l^2 20l - 84 0

Rearrange into standard form:

l^2 - 20l 84 0

Step 4: Using the Quadratic Formula

The quadratic formula is given by:

l [-b ± sqrt(b^2 - 4ac)] / (2a)

where:

a 1 b -20 c 84

Calculate the discriminant:

Δ b^2 - 4ac

Δ (-20)^2 - 4(1)(84)

Δ 400 - 336 64

Solving for l:

l [20 ± sqrt(64)] / 2

l [20 ± 8] / 2

This gives us two solutions for l:

l (20 8) / 2 14 l (20 - 8) / 2 6

Step 5: Finding Corresponding Widths

For each l, calculate the corresponding width (w 20 - l):

If l 14, w 20 - 14 6 If l 6, w 20 - 6 14

Thus, the possible dimensions of the rectangle are:

Length: 14 cm, Width: 6 cm Length: 6 cm, Width: 14 cm

Conclusion

By solving the equations, we have determined that the dimensions of the rectangle can be either 14 cm by 6 cm or 6 cm by 14 cm given the perimeter and area provided.