Solving Rectangle Problems: Finding Perimeter Given Area and Ratio
This article explains how to solve for the perimeter of a rectangle when the area and the ratio of its length to its width are given. It provides a step-by-step approach with examples to help understand the concepts better.
Introduction to Rectangle Problems
A rectangle is a basic geometric shape with four right angles and opposite sides of equal length. The perimeter of a rectangle is the total distance around it, while the area is the amount of space inside the rectangle. Solving for one of these values given the other along with a ratio can be achieved through a series of algebraic manipulations.
Example 1: Area of 300 cm2 and Length:Width Ratio of 4:3
The problem states that the area of a rectangle is 300 cm2 and the ratio of its length to its width is 4:3. The first step is to use the ratio to express the length and width in terms of a common variable.
Step-by-Step Solution
Step 1: Express the ratio in terms of a variable. Let the width be 3k and the length be 4k, where k is a constant.
Step 2: Use the area formula for a rectangle to set up an equation. The area of the rectangle is given by the product of its length and width.
Area length times; width 4k times; 3k 12k2
Step 3: Set the area equal to the given area and solve for k.
12k2 300 k2 300 / 12 25 k √25 5
Step 4: Substitute k back into the expressions for the length and width.
Length 4k 4 times; 5 20 cm Width 3k 3 times; 5 15 cm
Perimeter Calculation
The perimeter of a rectangle is the sum of all its sides, given by the formula:
Perimeter 2(length width) 2(20 15) 2(35) 70 cm
Example 2: Area of 252 cm2 and Length:Width Ratio of 9:7
In this second example, the area of a rectangle is 252 cm2 and the ratio of its length to its width is 9:7. The solution approach is similar to the previous example but uses different values.
Step-by-Step Solution
Step 1: Express the ratio in terms of a variable. Let the length be 9k and the width be 7k.
Step 2: Use the area formula to set up an equation.
Area length times; width 9k times; 7k 63k2
Step 3: Set the area equal to the given area and solve for k.
63k2 252 k2 252 / 63 4 k √4 2
Step 4: Substitute k back into the expressions for the length and width.
Length 9k 9 times; 2 18 cm Width 7k 7 times; 2 14 cm
Perimeter Calculation
Perimeter 2(length width) 2(18 14) 2(32) 64 cm
Conclusion
Understanding how to solve for the perimeter of a rectangle given its area and ratio of length to width is a fundamental skill in geometry. By expressing the length and width in terms of a common variable and using the area formula, the problem can be simplified and solved systematically.
Frequently Asked Questions
Q: How do you find the perimeter of a rectangle?
A: The formula for the perimeter of a rectangle is 2(length width). This is derived from the fact that opposite sides of a rectangle are equal.
Q: What is the relationship between length, width, and area?
A: The area of a rectangle is given by the product of its length and width (Area length times; width).
Q: How do you use the ratio to find the dimensions of a rectangle?
A: Given the ratio of length to width, a variable can be used to express both the length and width. By substituting these expressions into the area formula and solving for the variable, the actual dimensions can be found.