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 Context

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 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?

Key Insights

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

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Final Thoughts

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.

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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.