A rectangle has sides 9 cm and 6 cm. What is the perimeter
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Welcome to our lesson on calculating the perimeter of rectangles. The perimeter is the total distance around the outside of any shape. Think about walking around the edge of a garden or putting a fence around your yard - that's measuring perimeter. Today we'll learn how to calculate the perimeter of a rectangle with sides measuring 9 cm and 6 cm.
A rectangle is a four-sided shape with special properties that make perimeter calculation straightforward. All rectangles have opposite sides that are equal in length and all angles are right angles, measuring 90 degrees. This means if we know the length of two adjacent sides, we automatically know all four sides. In our example, we have a rectangle with one pair of sides measuring 9 cm and the other pair measuring 6 cm.
Let's visualize our rectangle with clear labels to understand what we're working with. The longer sides measure 9 centimeters each, and the shorter sides measure 6 centimeters each. Notice how opposite sides are equal - this is a defining characteristic of rectangles. By labeling all four sides, we can clearly see that we have two sides of 9 cm and two sides of 6 cm that make up our complete shape.
Mathematicians have developed a simple formula to calculate rectangle perimeter efficiently. The perimeter of a rectangle equals twice the length plus twice the width, written as P = 2l + 2w. This formula works because we're adding all four sides together - two lengths and two widths. We can also write this as P = 2(l + w), which is mathematically equivalent and sometimes easier to calculate.
Now let's identify which measurements correspond to length and width in our formula. We can assign the 9 cm measurement as our length (l) and the 6 cm measurement as our width (w). It's important to note that it doesn't matter which dimension we call length or width - the mathematical result will be the same. The key is being consistent with our assignments throughout the calculation process.
Let's substitute our values into the perimeter formula P = 2l + 2w. Replacing l with 9 cm and w with 6 cm, we get P = 2(9) + 2(6). This step is crucial because it translates our visual problem into a mathematical expression that we can solve systematically. Always remember to include the units in your substitution to maintain dimensional consistency.
Following the order of operations, we first perform the multiplication steps in our formula. Two times nine equals eighteen, and two times six equals twelve. This gives us P = 18 + 12. Breaking down the calculation into smaller steps helps prevent arithmetic errors and makes the process clearer. Each multiplication represents the total length of the two parallel sides of our rectangle.
The final step is to add our two products together: eighteen plus twelve equals thirty. Therefore, P = 30 centimeters. This sum represents the total distance around our rectangle - the perimeter we've been seeking. The unit of measurement remains centimeters because we're measuring a one-dimensional distance around the two-dimensional shape.
Let's verify our answer using an alternative method - adding all four sides individually. We have 9 + 6 + 9 + 6 centimeters. Adding these step by step: nine plus six equals fifteen, fifteen plus nine equals twenty-four, and twenty-four plus six equals thirty centimeters. This confirms our previous result and demonstrates that both methods yield the same correct answer.
Our calculated perimeter of 30 centimeters represents the total linear distance around the rectangle's boundary. If you were to walk along the edge of this rectangle, you would travel exactly 30 centimeters to return to your starting point. This measurement is useful for practical applications like determining how much fencing is needed to enclose a rectangular area or how much trim is required for a rectangular picture frame.
Let's double-check our answer using logical reasoning. Since we have two sides of 9 cm each, that's 18 cm total for the long sides. The two shorter sides of 6 cm each contribute 12 cm. Eighteen plus twelve indeed equals 30 cm, confirming our calculation. This verification step is important in mathematics to ensure our work is accurate and our reasoning is sound.
Understanding how to calculate perimeter has numerous real-world applications in daily life. Architects use perimeter calculations when designing building foundations, gardeners use them when planning fence installations, and crafters use them when creating picture frames or quilt borders. The ability to quickly and accurately calculate perimeter is a fundamental mathematical skill that proves valuable in many practical situations.