Plywood Calculation For 13 Isosceles Right Triangles

Ariana faces a fascinating geometric challenge: determining the amount of plywood required to cut 13 isosceles right triangles, each with a hypotenuse of 60 cm. This problem combines the principles of geometry, specifically the properties of right triangles and the Pythagorean theorem, with practical application in material estimation. To solve this, we need to delve into the characteristics of isosceles right triangles, calculate the length of their legs, find the area of a single triangle, and then multiply that by the number of triangles Ariana intends to cut. This article will walk through the step-by-step process, providing a comprehensive understanding of the calculations involved and ensuring accurate material estimation for Ariana's project.

Understanding Isosceles Right Triangles

To accurately calculate the plywood needed, it's crucial to first understand the properties of an isosceles right triangle. An isosceles right triangle, also known as a 45-45-90 triangle, possesses unique characteristics that are essential for solving geometric problems. By definition, an isosceles triangle has two sides of equal length, and in the case of a right triangle, these two sides are the legs, which form the right angle. The angle opposite the right angle is called the hypotenuse, which is the longest side of the triangle. The angles opposite the two equal sides are also equal, each measuring 45 degrees, while the right angle measures 90 degrees. This specific angle configuration gives rise to the name 45-45-90 triangle, making it a special type of triangle with distinct properties.

Key Properties of Isosceles Right Triangles

One of the most important properties of an isosceles right triangle is the relationship between its sides. This relationship is described by the Pythagorean theorem, a fundamental concept in geometry. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). Mathematically, this is expressed as a² + b² = c², where a and b are the lengths of the legs, and c is the length of the hypotenuse. In an isosceles right triangle, the two legs are of equal length, so we can denote them both as 'a'. Thus, the Pythagorean theorem for an isosceles right triangle simplifies to a² + a² = c², which further simplifies to 2a² = c². This equation is the cornerstone for calculating the lengths of the legs when the hypotenuse is known, as in Ariana's problem.

Applying the Pythagorean Theorem

In the context of Ariana's project, we know that the hypotenuse of each triangle is 60 cm. Our objective is to determine the length of the legs of the triangle, which will allow us to calculate the area of each triangle. By applying the Pythagorean theorem, we can set up the equation 2a² = 60². Solving this equation for 'a' will give us the length of each leg. This is a critical step in finding the area because the area of a right triangle is calculated as half the product of the lengths of its legs. Understanding and applying the Pythagorean theorem correctly is therefore essential for accurately estimating the amount of plywood needed for Ariana's project. The properties of isosceles right triangles and the Pythagorean theorem are not just abstract mathematical concepts but practical tools that can be applied to real-world problems, such as this one, highlighting the importance of geometry in everyday life.

Calculating the Leg Length

To determine the amount of plywood Ariana will need, the first critical step is calculating the length of the legs of the isosceles right triangles. As established earlier, an isosceles right triangle has two equal legs, and we can use the Pythagorean theorem to find their lengths given the hypotenuse. In this case, the hypotenuse is 60 cm. The Pythagorean theorem, in the context of an isosceles right triangle, simplifies to 2a² = c², where 'a' represents the length of each leg and 'c' represents the length of the hypotenuse. This equation is our key to unlocking the length of the legs.

Applying the Formula

Substituting the known value of the hypotenuse (c = 60 cm) into the equation, we get 2a² = 60². This equation now becomes a simple algebraic problem that we can solve for 'a'. First, we calculate 60² which equals 3600. So, the equation becomes 2a² = 3600. To isolate a², we divide both sides of the equation by 2, resulting in a² = 1800. This step simplifies the equation and brings us closer to finding the value of 'a', the length of each leg.

Solving for 'a'

The final step in calculating the leg length involves taking the square root of both sides of the equation a² = 1800. This will give us the value of 'a', which is the length of each leg of the triangle. The square root of 1800 is approximately 42.43 cm. Therefore, each leg of the isosceles right triangles that Ariana intends to cut is approximately 42.43 cm long. This calculation is crucial because the length of the legs is needed to determine the area of each triangle, which in turn will allow us to calculate the total amount of plywood required for all 13 triangles. Accurately calculating the leg length ensures that the subsequent area calculations and material estimation are precise, saving both time and resources in the project. Understanding and applying these mathematical principles is fundamental to successfully completing practical tasks such as this one.

Determining the Area of One Triangle

Once we have the length of the legs, calculating the area of one isosceles right triangle becomes a straightforward process. The area of any triangle is generally calculated as half the product of its base and height. However, in the special case of a right triangle, the two legs serve as the base and height, simplifying the calculation. Therefore, for an isosceles right triangle, the area can be found by taking half the product of the lengths of its two equal legs. This method provides an efficient and accurate way to determine the area of the triangles in Ariana's project.

Applying the Area Formula

In the previous section, we calculated the length of each leg of the triangle to be approximately 42.43 cm. To find the area of one triangle, we will use the formula: Area = 0.5 * base * height. Since the base and height are both the legs of the triangle, they are equal in length. Substituting the value of the leg length into the formula, we get Area = 0.5 * 42.43 cm * 42.43 cm. This calculation involves multiplying the leg length by itself and then halving the result.

Calculating the Area

Performing the multiplication, 42.43 cm * 42.43 cm equals approximately 1800.3049 cm². Now, we multiply this result by 0.5 (or divide by 2) to get the area of one triangle. So, Area = 0.5 * 1800.3049 cm² which equals approximately 900.1525 cm². Therefore, the area of one isosceles right triangle with legs of approximately 42.43 cm is about 900.1525 square centimeters. This value is a crucial piece of information as it represents the amount of plywood needed for a single triangle. With this area calculated, we can now proceed to determine the total amount of plywood required for all 13 triangles that Ariana intends to cut. The accurate determination of the area of a single triangle is essential for the overall material estimation and ensures that Ariana purchases the correct amount of plywood for her project.

Calculating the Total Plywood Needed

Now that we have the area of a single isosceles right triangle, the final step in determining the amount of plywood Ariana needs is to calculate the total area required for all 13 triangles. This is a simple multiplication problem, where we multiply the area of one triangle by the total number of triangles. This step provides the total surface area of plywood that Ariana will need to cut out all the triangles, ensuring that she has enough material for her project.

Multiplying the Area by the Number of Triangles

We previously calculated the area of one triangle to be approximately 900.1525 cm². Ariana wants to cut 13 such triangles. To find the total area, we multiply these two values together: Total Area = Area of one triangle * Number of triangles. Substituting the values, we get Total Area = 900.1525 cm² * 13. This calculation will give us the total area of plywood in square centimeters that Ariana will need.

Calculating the Total Area

Performing the multiplication, 900.1525 cm² multiplied by 13 equals approximately 11701.9825 cm². Therefore, Ariana will need approximately 11701.9825 square centimeters of plywood to cut out all 13 isosceles right triangles. This value represents the minimum amount of plywood required, assuming no waste during the cutting process. In practical scenarios, it is often wise to add a small percentage to account for waste due to cuts and imperfections in the plywood sheet. However, this calculation provides a solid estimate of the plywood needed for the project. This final calculation completes the process of determining the material requirements for Ariana's project, demonstrating how geometric principles and calculations can be applied to solve real-world problems.

Conclusion

In conclusion, determining the amount of plywood Ariana needs involves a methodical application of geometric principles and calculations. Starting with an understanding of the properties of isosceles right triangles and the Pythagorean theorem, we successfully calculated the leg length of each triangle. This leg length was then used to find the area of a single triangle, and finally, the total area required for all 13 triangles was determined. The process demonstrates the practical application of mathematical concepts in everyday situations. Ariana will need approximately 11701.9825 square centimeters of plywood for her project. By following these steps, anyone can accurately estimate material requirements for similar geometric projects, ensuring efficiency and cost-effectiveness in their endeavors. This exercise not only provides a solution to a specific problem but also reinforces the value of mathematical thinking in problem-solving.