Question

9 practice problems
6 from unit 1, lesson 3
here is an octagon. (note: the diagonal sides of the octagon are not 4 inches long.)
a. while estimating the area of the octagon, lin reasoned that it must be less than 100 square inches. do you agree? explain your reasoning.
b. find the exact area of the octagon. show your reasoning.
learning targets
+ i can use the area formula to find the area of any triangle.
+ i can write and explain the formula for the area of a triangle.
+ i know what the terms \base\ and \height\ refer to in a triangle.

Explanation:

Step1: Find the area of the outer - rectangle

The outer - rectangle has length \(l = 3 + 4+3=10\) inches and width \(w = 3 + 4+3 = 10\) inches. The area of a rectangle is \(A_{rect}=l\times w\). So, \(A_{rect}=10\times10 = 100\) square inches. Since the octagon is inside the rectangle, its area must be less than the area of the rectangle. So, Lin is correct.

Step2: Divide the octagon into smaller parts

We can divide the octagon into a rectangle in the middle and 4 right - angled triangles at the corners. The middle rectangle has length \(l = 4\) inches and width \(w = 4\) inches. So, the area of the middle rectangle \(A_{1}=4\times4=16\) square inches.

Step3: Calculate the area of one right - angled triangle

Each of the 4 right - angled triangles has base \(b = 3\) inches and height \(h = 3\) inches. The area of a right - angled triangle is \(A_{triangle}=\frac{1}{2}\times b\times h\). So, \(A_{triangle}=\frac{1}{2}\times3\times3=\frac{9}{2}\) square inches.

Step4: Calculate the total area of the 4 right - angled triangles

The total area of the 4 right - angled triangles is \(A_{2}=4\times\frac{9}{2}=18\) square inches.

Step5: Calculate the area of the octagon

The area of the octagon \(A = A_{1}+A_{2}\). Substituting the values of \(A_{1}\) and \(A_{2}\), we get \(A=16 + 18=34\) square inches.

Answer:

a. Yes, Lin is correct. The octagon is inscribed in a 10 - inch by 10 - inch rectangle with an area of 100 square inches, so the area of the octagon is less than 100 square inches.
b. 34 square inches. The area is found by adding the area of the middle rectangle (\(4\times4 = 16\) square inches) and the area of the 4 right - angled triangles (\(4\times\frac{1}{2}\times3\times3=18\) square inches).