The ratio of the lengths of the sides of a rectangle is 3:4. If the perimeter is 56 cm, what is the area of the rectangle?
Title: How to Calculate the Area of a Rectangle with a 3:4 Side Ratio and Perimeter of 56 cm
If you're given that the sides of a rectangle are in a 3:4 ratio and its perimeter is 56 cm, you might wonder: What is the area of this rectangle? This SEO-optimized guide walks you through the step-by-step solution, helping you master similar geometry problems efficiently.
Understanding the 3:4 Side Ratio
A rectangle with side lengths in the ratio 3:4 can be represented using a variable multiplier. Let the shorter side be 3x and the longer side 4x, where x is a positive real number. This notation ensures that the proportions always match 3 to 4 regardless of x.
Using the Perimeter to Solve for x
The formula for the perimeter (P) of a rectangle is: P = 2 × (length + width) Substituting the expressions based on the ratio: 56 = 2 × (3x + 4x) Simplify inside the parentheses: 56 = 2 × (7x) 56 = 14x Now solve for x: x = 56 ÷ 14 = 4
Finding the Actual Side Lengths
Now plug x = 4 back into the expressions for the sides:
- Shorter side = 3x = 3 × 4 = 12 cm
- Longer side = 4x = 4 × 4 = 16 cm
Calculating the Area
The area (A) of a rectangle is: A = length × width So, A = 12 × 16 = 192 cm²
Final Answer
A rectangle with a 3:4 side ratio and a perimeter of 56 cm has an area of 192 square centimeters. This structured approach—using ratios, algebra, and basic formulas—enables quick and accurate calculations for similar problems.
Key SEO Keywords: rectangle area calculation 3:4 ratio perimeter, find rectangle area with 3-4 sides, how to calculate rectangle area given ratio and perimeter, rectangle perimeter and area problem solving
Meta Description: Learn how to calculate the area of a rectangle with a 3:4 side ratio and a perimeter of 56 cm. Step-by-step solution with formulas and real-life geometry examples. Perfect for students and math learners.
