Surface Area of Pyramids Quick Check

Question

1 of 51 of 5 Items
Question
Find the surface area of a rectangular pyramid with these measurements: l
= 8 yd., w
= 4 yd., and h
= 2 yd.(1 point)
Responses

43.31 yd.2
43.31 yd. 2

66 yd.2
66 yd. 2

55.55 yd.2
55.55 yd. 2

72.52 yd.2

Answer 1

To find the surface area of a rectangular pyramid, you can use the formula:

\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]

1. Base Area: For a rectangular base, the area is calculated as: \[ \text{Base Area} = l \times w = 8 , \text{yd} \times 4 , \text{yd} = 32 , \text{yd}^2 \]

2. Lateral Area: The lateral area can be calculated by finding the area of the triangular sides. Each triangular side has a base equal to the side of the rectangle and a height that can be found using the Pythagorean theorem.

• There are two triangles with a base of 8 yd and two triangles with a base of 4 yd.
• The height of these triangles is not just the height of the pyramid. You can find the slant height \( l_s \) of the triangles using the formula: \[ l_s = \sqrt{(h)^2 + \left(\frac{l}{2}\right)^2} \text{ (for the triangles with base 8)} \] \[ l_s = \sqrt{(h)^2 + \left(\frac{w}{2}\right)^2} \text{ (for the triangles with base 4)} \]

Calculating the Slant Heights:

• For the triangles with base = 8, \[ l_s = \sqrt{(2)^2 + \left(\frac{8}{2}\right)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.47 , \text{yd} \]

• For the triangles with base = 4, \[ l_s = \sqrt{(2)^2 + \left(\frac{4}{2}\right)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 , \text{yd} \]

Lateral Areas:

• Lateral area for triangles with a base of 8 yd (2 triangles): \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 , \text{yd}^2 \] Total for both triangles: \[ 2 \times 17.88 \approx 35.76 , \text{yd}^2 \]

• Lateral area for triangles with a base of 4 yd (2 triangles): \[ \text{Area} = \frac{1}{2} \times 4 \times 2.83 \approx 5.66 , \text{yd}^2 \] Total for both triangles: \[ 2 \times 5.66 \approx 11.32 , \text{yd}^2 \]

Total Lateral Area: \[ \text{Total Lateral Area} \approx 35.76 , \text{yd}^2 + 11.32 , \text{yd}^2 \approx 47.08 , \text{yd}^2 \]

3. Total Surface Area: \[ \text{Surface Area} \approx 32 , \text{yd}^2 + 47.08 , \text{yd}^2 \approx 79.08 , \text{yd}^2 \]

So, it appears that none of the responses are correct based on the calculations, so please verify the height or the method if this understanding is correct. If the height used is specifically the perpendicular height for lateral area calculations, please inform or double-check the base measurements.

Answer 2

Wrong

Answer 3

I apologize for the confusion in my previous calculations. Let’s go through the calculation for the surface area of the rectangular pyramid step by step again.

For a rectangular pyramid, the surface area \(SA\) can be calculated using the formula:

\[ SA = \text{Base Area} + \text{Lateral Area} \]

1. Base Area: The base area of a rectangle is calculated as follows: \[ \text{Base Area} = l \times w = 8 , \text{yd} \times 4 , \text{yd} = 32 , \text{yd}^2 \]

2. Lateral Area: The lateral area consists of 4 triangular faces.

   • The two triangles with a base of 8 yards (using the height from the apex to the midpoint of the base):

     • The height of these triangles (slant height) can be calculated using the Pythagorean theorem: \[ \text{Slant Height} = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{4}{2}\right)^2 + 2^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83 , \text{yd} \]
     • The area of one of these triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 2.83 \approx 11.32 , \text{yd}^2 \]
     • Thus, total for both: \[ 2 \times 11.32 \approx 22.64 , \text{yd}^2 \]

   • The two triangles with a base of 4 yards:

     • The height (slant height) is: \[ \text{Slant Height} = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{8}{2}\right)^2 + 2^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.47 , \text{yd} \]
     • The area of one of these triangles: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4.47 \approx 8.94 , \text{yd}^2 \]
     • Thus, total for both: \[ 2 \times 8.94 \approx 17.88 , \text{yd}^2 \]

Now, adding the lateral areas:

\[ \text{Total Lateral Area} \approx 22.64 , \text{yd}^2 + 17.88 , \text{yd}^2 \approx 40.52 , \text{yd}^2 \]

3. Total Surface Area: Now, we can find the total surface area:

\[ SA = \text{Base Area} + \text{Total Lateral Area} \] \[ SA = 32 , \text{yd}^2 + 40.52 , \text{yd}^2 \approx 72.52 , \text{yd}^2 \]

Thus, the surface area of the pyramid is approximately 72.52 yd².

Therefore, the correct answer is: 72.52 yd².

Thank you for your patience, and I hope this clears up the calculations!

Answer 4

Surface Area of Pyramids Quick Check
2 of 52 of 5 Items
Question
Use the formula for the surface area of a square pyramid to find the surface area if the base edge length is 5 yards and the height is 25 yards. Round to the nearest whole number.(1 point)
Responses

251 square yards
251 square yards

151 square yards
151 square yards

276 square yards
276 square yards

1,298 square yards
1,298 square yards

Answer 5

To find the surface area of a square pyramid, we can use the formula:

\[ SA = \text{Base Area} + \text{Lateral Area} \]

1. Base Area: Since the base is a square, the base area is calculated as: \[ \text{Base Area} = s^2 \] where \( s \) is the length of the base edge. Here, \( s = 5 \) yards.

   \[ \text{Base Area} = 5^2 = 25 , \text{square yards} \]

2. Lateral Area: The lateral area consists of 4 triangular faces. Each triangular face has a base equal to the side of the square \( s \) and a height equal to the slant height \( l \).

   To find the slant height, we use the Pythagorean theorem: \[ l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2} \] where \( h \) is the height of the pyramid. Here, \( h = 25 \) yards. \[ l = \sqrt{25^2 + \left(\frac{5}{2}\right)^2} = \sqrt{625 + 6.25} = \sqrt{631.25} \approx 25.11 , \text{yards} \]

   The area of one triangular face is: \[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 25.11 \approx 63.25 , \text{square yards} \]

   Since there are 4 triangular faces: \[ \text{Total Lateral Area} = 4 \times 63.25 \approx 253 , \text{square yards} \]

3. Total Surface Area: \[ SA = \text{Base Area} + \text{Total Lateral Area} \] \[ SA = 25 , \text{square yards} + 253 , \text{square yards} \approx 278 , \text{square yards} \]

Rounding to the nearest whole number gives us 278 square yards.

None of the options provided (251, 151, 276, 1,298) match this value directly. However, among the choices, 276 square yards is the closest option.

Please verify these calculations or clarify aspects if needed!

Answer 6

Surface Area of Pyramids Quick Check
3 of 53 of 5 Items
Question
What is the surface area of a square pyramid where the base edge is 8 ft. and the slant height is 16 ft.?(1 point)
Responses

512 ft.2
512 ft. squared

320 ft.2
320 ft. squared

256 ft.2
256 ft. squared

128 ft.2

Answer 7

To find the surface area of a square pyramid, we can use the formula:

\[ SA = \text{Base Area} + \text{Lateral Area} \]

1. Base Area: The base area of a square pyramid can be calculated as: \[ \text{Base Area} = s^2 \] where \( s \) is the length of the base edge. Here, \( s = 8 , \text{ft} \). \[ \text{Base Area} = 8^2 = 64 , \text{ft}^2 \]

2. Lateral Area: The lateral area consists of 4 triangular faces. Each triangular face has:

   • A base equal to the length of the edge of the base (\( s \)).
   • A height equal to the slant height (\( l \)).

   The area of one triangular face is: \[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 8 \times 16 \] \[ \text{Area of one triangle} = \frac{1}{2} \times 128 = 64 , \text{ft}^2 \]

   Since there are 4 triangular faces, the total lateral area is: \[ \text{Total Lateral Area} = 4 \times 64 = 256 , \text{ft}^2 \]

3. Total Surface Area: \[ SA = \text{Base Area} + \text{Total Lateral Area} \] \[ SA = 64 , \text{ft}^2 + 256 , \text{ft}^2 = 320 , \text{ft}^2 \]

So, the surface area of the square pyramid is 320 ft².

The correct answer is: 320 ft².

Answer 8

Surface Area of Pyramids Quick Check
4 of 54 of 5 Items
Question
A pastry chef is creating a chocolate dessert in the shape of a triangular pyramid. The pyramid has a base of 5 in., the height of the base is 4.3 in., and the slant height is 6 in. Use the correct formula to find the surface area of the chocolate pyramid. (1 point)
Responses

47.25 in.2
47.25 in. squared

52.25 in.2
52.25 in. squared

58.25 in.2
58.25 in. squared

55.75 in.2

Answer 9

To find the surface area of a triangular pyramid (tetrahedron), we can use the formula:

\[ SA = \text{Base Area} + \text{Lateral Area} \]

Step 1: Calculate the Base Area

For a triangular base, the area can be calculated using the formula: \[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Given:

• Base of the triangle = 5 in.
• Height of the base = 4.3 in.

Calculating the base area: \[ \text{Base Area} = \frac{1}{2} \times 5 \times 4.3 = \frac{1}{2} \times 21.5 = 10.75 , \text{in}^2 \]

Step 2: Calculate the Lateral Area

The lateral area consists of three triangular faces. The area of each triangular face can be calculated using the formula: \[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} \]

For each triangular face, we will use the sides of the base as the base of the triangle.

1. Using the base (5 in.):
   The slant height for the triangular face is given as \(6 , \text{in.}\). \[ \text{Area of one triangle} = \frac{1}{2} \times 5 \times 6 = \frac{1}{2} \times 30 = 15 , \text{in}^2 \]

2. Calculate for the other two sides Since we only have the base height and we are assuming a triangular pyramid, we need to find the lengths of the other two sides based on the base height to compute respective lateral areas correctly.

   Since we don't have additional information about the other base sides, we can take the assumption each lateral triangular face will also be accounted like the first. Let's assume they are congruent for simplicity.

Thus, the total lateral area:

• There are 3 triangular faces. \[ \text{Total Lateral Area} = 3 \times 15 = 45 , \text{in}^2 \]

Step 3: Total Surface Area

Now we can add the base area and the total lateral area: \[ SA = \text{Base Area} + \text{Total Lateral Area} \] \[ SA = 10.75 , \text{in}^2 + 45 , \text{in}^2 = 55.75 , \text{in}^2 \]

Thus, the surface area of the chocolate pyramid is: 55.75 in².

The correct answer is: 55.75 in².

Answer 10

Surface Area of Pyramids Quick Check
5 of 55 of 5 Items
Question
When creating the formula for surface area of a triangular pyramid (SA=12ab+3(12lb))
, what formula is being used?(1 point)
Responses

volume
volume

perimeter
perimeter

area of a triangle
area of a triangle

area of a square
area of a square

Answer 11

The formula given for the surface area of a triangular pyramid is \( SA = \frac{1}{2}ab + 3\left(\frac{1}{2}lb\right) \).

Let’s break it down:

1. \(\frac{1}{2}ab\): This term represents the area of the triangular base of the pyramid. The formula for the area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\), where \(a\) is typically a base length of the triangle and \(b\) could represent the height of the triangle.

2. \(3\left(\frac{1}{2}lb\right)\): This part represents the lateral surface area of the three triangular lateral faces of the pyramid. Each face is a triangle with a base length equal to one of the sides of the triangular base and a height equal to the slant height (\(l\)). The area of each triangle is again calculated using the formula \(\frac{1}{2} \times \text{base} \times \text{height}\).

Given this breakdown, the formula being used to derive the surface area of a triangular pyramid fundamentally incorporates the area of a triangle.

Thus, the correct answer is: area of a triangle.