Problem 47 - Algebra and Trigonometry

Question

Completed in 1902 in New York City, the Flatiron Building is triangular shaped and bounded by 22nd Street, Broadway, and 5th Avenue. The building measure approximately 87 feet on the 22 nd Street side, 190 feet on the Broadway side, and 173 feet on the 5 th Avenue side. Approximate the ground area covered by the building.

Step-by-Step Solution

Answer

Approximate area covered by the Flatiron Building is 7510 square feet.

Step 1 - Identify the sides of the triangle

Label the sides of the triangle with the given lengths. Let side a = 87 feet, side b = 190 feet, and side c = 173 feet.

Step 2 - Use Heron's Formula for the area of a triangle

Heron's formula allows for finding the area when all three sides of the triangle are known. The formula is: \[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \], where \[ s = \frac{a + b + c}{2} \].

Step 3 - Calculate the semi-perimeter (s)

Compute the semi-perimeter (s): \[ s = \frac{87 + 190 + 173}{2} = 225 \] feet.

Step 4 - Apply the semi-perimeter to Heron's formula

Substitute the known values into Heron's formula: \[ \text{Area} = \sqrt{225(225-87)(225-190)(225-173)} \].

Step 5 - Simplify inside the square root

First, perform the subtractions inside the parentheses: \[ (225-87) = 138 \], \[ (225-190) = 35 \] and \[ (225-173) = 52 \]. Now we have: \[ \text{Area} = \sqrt{225 \cdot 138 \cdot 35 \cdot 52} \].

Step 6 - Perform the multiplications

Multiply the numbers inside the square root: \[ 225 \cdot 138 = 31050 \], \[ 31050 \cdot 35 = 1086750 \] and \[ 1086750 \cdot 52 = 56565000 \]. Now we have: \[ \text{Area} = \sqrt{56565000} \].

Step 7 - Calculate the square root

Finally, find the square root of 56565000 to approximate the area: \[ \text{Area} \approx 7510 \text{ square feet} \].

Key Concepts

Triangle Area Calculation

Understanding how to calculate the area of a triangle is important, especially when you know the lengths of all three sides. One common method used for this purpose is Heron's Formula. This formula is particularly useful for non-right-angled triangles.
Heron's Formula states that for any triangle with sides labeled as a, b, and c, the area can be calculated using the formula: \[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \] where s is the semi-perimeter of the triangle. The semi-perimeter is half of the perimeter and calculated by summing all three sides and then dividing by 2: \[ s = \frac{a + b + c}{2} \] If the sides of the triangle are a = 87 feet, b = 190 feet, and c = 173 feet, the semi-perimeter (s) can be calculated first. Then, these values can be substituted into Heron's Formula to find the area.
Using this method ensures accuracy when the triangle is not a simple shape like a right triangle, which makes the area calculation straightforward and reliable.

Semi-perimeter

The concept of semi-perimeter is a key part of Heron's Formula. It first involves computing the perimeter of the triangle, which is simply the sum of the lengths of all its sides: \[ \text{Perimeter} = a + b + c \]. The semi-perimeter is half of the perimeter, simplifying the equation involved in Heron's Formula.
Let's break it down:

• Identify the lengths of the sides. For the Flatiron Building, we have a = 87 feet, b = 190 feet, and c = 173 feet.
• Add these lengths to get the perimeter: 87 + 190 + 173 = 450 feet.
• Divide the perimeter by 2 to get the semi-perimeter: s = 225 feet.

This value of s will be used in further calculations to find the area using Heron's Formula. Knowing the semi-perimeter is crucial since it helps in breaking down the complex area calculation into simpler steps.

Square Root Calculation

Finally, we need to interpret the numbers inside the square root in Heron's Formula. This step ensures that we reach an appropriate value for the area. Once you have calculated the semi-perimeter and substituted all values into Heron's Formula, you'll generally have to perform several multiplication operations inside the square root.
For example, substituting into the formula for the Flatiron Building, you get: \[ \text{Area} = \sqrt{225 \cdot 138 \cdot 35 \cdot 52} \]. These values need to be multiplied step-by-step:

• First, multiply 225 by 138 resulting in 31050.
• Next, multiply 31050 by 35 to get 1086750.
• Then, multiply 1086750 by 52 to achieve 56565000.

Now, the area is the square root of 56565000. Using a calculator, find the square root to approximate the area: \[ \text{Area} \approx 7510 \text{ square feet} \].
Understanding how to perform these multiplications and square root calculations accurately ensures that you get the correct and precise area for any triangle, even complex shapes like the Flatiron Building.