Use the diagram below for questions 25-30. The barn is shaped like a pentagonal prism with dimensions shown in feet.

Question

1. What is the width of the roof? (HINT: Use the Pythagorean Theorem)
2. What is the area of the roof? (Both sides)
3. What is the floor area of the barn?
4. What is the area of the rectangular sides of the barn?
5. What is the area of the two pentagon sides of the barn? (HINT: Find the area of two congruent trapezoids for each side)
6. Find the total surface area of the barn (Roof and sides).

The diagram is accessed through the link https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A6bdd7ed2d3f3b5385340d436a1e13a1032d38985199c98bb6024cb33%2BIMAGE_TINY%2BIMAGE_TINY.1

Answer 1

1. To find the width of the roof, use the Pythagorean theorem where c is the width, a is one half the width of the barn, and b is the height from the eaves to the top of the roof. The diagram shows that the height is congruent with the height from the base to the eaves, given as 20. So we know b = 20. The width of the barn is 42, and one half of that , or 21, is a. Plug these into the Pythagorean theorem and the width of the roof is 28.65.
2. For the area of the roof, multiply the width of the roof (28.65) times the length (70) and double that because there are two parts to the roof. 2 x 28.65 x 70 = 4011.
3. The floor area is base times length of the barn, 42 x 70 = 2940.
4. The total area of the rectangular portions of the sides of the barn would be that portion below the eaves. The height of the barn is 20. The front and back are each 70 long, so 2 x 20 x 70 = 2800 is the total of those surfaces. The sides are 42 long and also 20 high, so 2 x 42 x 20 = 1680 is the surface area of the sides. Add those together for the total, 2800 +1680 = 4480.
5. For the two pentagon sides, an easy method would be to break the sides down into componet polygons of a rectangle topped by a triangle. The rectangular portion of each side is the width of the barn times the height up to the eaves, or 42 x 20 = 840. That is topped by a triangle that has a base of 42 and a height of 20 (remember those lines that denote congruence) so using the formula for the area of a triangle, 1/2 x b x h, we have 1/2 x 42 x 20 = 420. That means each pentagonal side is 840 + 420 = 1260, and there are two of them so the total is 2520.
6. For the total area of the barn, roof and sides (not the floor) we need to add the two pentagonal sides, the two rectangular sides, ant the two roof portions. From Q5, we know the pentagonal sides add up to 2560. The barn is 70 long and 20 high to the eaves, so each side is 20 x 70 = 1400, and the total for the front and back sides is 2800. From Q2 we determined that the area of the roof is 4011. So add those together, 4011 + 2800 + 2560 = 9371.

Hope that helps.