Find the lateral area of a regular pyramid whose base is a square, whose slant height is 5 m, whose height is 3 m.
I think that the answer is 60. I subtracted 3 squared from 5 squared and I got 4. I multipled 4 by 2 and I got 8. I made a formula 3(1/2(8 x 5)) and I got 60 as my lateral area. Is this right?
If the slant height is 5 and the actual height it 3, then it is 4 m from the center to the middle of any of the squate sides. A side view would be two 3-4-5 right triangles, back to back.
The base length is therefore 8 m. Each trangular side has an area of
Aside = (1/2)8*5 = 20 m^2. The total lateral area (4 sides) would therefore be 80 m^2.
I think you mistake was assuming that there are three triangular sides. There are four
Yes, your calculation for finding the lateral area of the regular pyramid is correct. The formula for the lateral area of a regular pyramid is given by:
Lateral Area = (1/2) * perimeter of base * slant height
Since the base of the pyramid is a square, all four sides are equal in length. The perimeter of the square base can be calculated by multiplying the length of one side by 4.
Perimeter of square base = 4 * side length
In this case, the slant height is given as 5 m, and the height is given as 3 m. By applying the Pythagorean Theorem, you correctly found that the length of one side of the square base is 4 m (by subtracting 3 squared from 5 squared and taking the square root).
Substituting the values into the formula:
Lateral Area = (1/2) * (4 * 4) * 5
Lateral Area = (1/2) * 16 * 5
Lateral Area = 8 * 5
Lateral Area = 40 square meters
Therefore, the lateral area of the regular pyramid is indeed 40 square meters, not 60.
Yes, your answer of 60 for the lateral area of the regular pyramid is correct. Let's go through the process of calculating it step by step to understand how you arrived at the answer.
To find the lateral area of a regular pyramid, you can use the formula:
Lateral Area = (1/2) * perimeter of the base * slant height
In this case, the base of the pyramid is a square, so its perimeter can be found by multiplying the length of one side by 4.
The slant height of the pyramid is given as 5 m.
First, let's calculate the perimeter of the base. Since it is a square, all four sides are equal in length. However, the length of one side isn't provided in the question. Therefore, you'll need to find the length of one side based on the given information.
The height of the pyramid is given as 3 m. Since the pyramid is regular, the height forms a right triangle with half of the base's side length (which we will call 's'). The slant height forms the hypotenuse of this right triangle.
Applying the Pythagorean theorem, we have:
s^2 + (1/2s)^2 = slant height^2
s^2 + (1/4)s^2 = 5^2
s^2 + (1/4)s^2 = 25
Combining like terms:
(5/4)s^2 = 25
To solve for s, divide both sides by (5/4):
s^2 = 25 / (5/4)
s^2 = 20
Taking the square root of both sides:
s = sqrt(20)
Now that we have found the length of one side of the base (s), we can proceed to calculate the perimeter of the base.
Perimeter of the base = 4 * s
Perimeter of the base = 4 * sqrt(20)
Perimeter of the base = 8 * sqrt(5)
Now, we can substitute the values we have found into the lateral area formula:
Lateral Area = (1/2) * perimeter of the base * slant height
Lateral Area = (1/2) * (8 * sqrt(5)) * 5
Lateral Area = 4 * 5 * sqrt(5)
Lateral Area = 20 * sqrt(5)
Lateral Area ≈ 44.7 m² (rounded to one decimal place)
Therefore, the correct answer for the lateral area of the regular pyramid is approximately 44.7 square meters, which is different from your initial answer of 60.