A house shaped like an octagon has a large porch, as shown. What is the area of the house and the porch? 5 feet 5 feet 5 feet 5 feet House 10 feet 15 feet Porch 30 feet
The correct answer and explanation is:
To find the total area of the house and the porch, we must understand the shape and the given measurements. From the description:
- The house is shaped like a regular octagon, and
- The porch is a rectangular extension attached to one side of the house.
Step 1: Area of the House (Regular Octagon)
The house is a regular octagon with side length = 5 feet.
The formula to find the area AA of a regular octagon with side length ss is: A=2(1+2)s2A = 2(1 + \sqrt{2})s^2
Substitute s=5s = 5: A=2(1+2)(5)2=2(1+2)(25)A = 2(1 + \sqrt{2})(5)^2 = 2(1 + \sqrt{2})(25)
First, calculate the expression in steps:
- 2≈1.414\sqrt{2} \approx 1.414
- 1+2≈2.4141 + \sqrt{2} \approx 2.414
- 2×2.414=4.8282 \times 2.414 = 4.828
- 4.828×25=120.74.828 \times 25 = 120.7
So, the area of the house is approximately 120.7 square feet.
Step 2: Area of the Porch
From the description and measurements:
- The porch is a rectangle, measuring 10 feet by 15 feet.
Area of porch=10×15=150 square feet\text{Area of porch} = 10 \times 15 = 150 \text{ square feet}
Step 3: Total Area
Now, add both areas: Total area=House area+Porch area=120.7+150=270.7 square feet\text{Total area} = \text{House area} + \text{Porch area} = 120.7 + 150 = \boxed{270.7 \text{ square feet}}
Final Answer:
270.7 square feet\boxed{270.7 \text{ square feet}}
Explanation (300 words):
To determine the area of a house shaped like an octagon with an attached porch, we must calculate the area of two geometric shapes: a regular octagon (the house) and a rectangle (the porch).
A regular octagon has eight equal sides and eight equal angles. When calculating its area, a special formula is used: A=2(1+2)s2A = 2(1 + \sqrt{2})s^2
where ss is the side length. In this case, each side of the octagon is 5 feet. Plugging in the value gives us: A=2(1+2)(25)≈4.828×25=120.7 square feetA = 2(1 + \sqrt{2})(25) \approx 4.828 \times 25 = 120.7 \text{ square feet}
Next, we consider the porch, which is rectangular. The area of a rectangle is calculated using: Area=length×width\text{Area} = \text{length} \times \text{width}
With a given length of 10 feet and width of 15 feet, the area of the porch is: 10×15=150 square feet10 \times 15 = 150 \text{ square feet}
To find the total area, we add both parts: 120.7+150=270.7 square feet120.7 + 150 = 270.7 \text{ square feet}
This approach ensures accuracy by treating each shape separately, applying appropriate area formulas, and then combining their areas.