Find the Lateral area in cm square of the following right pryramid.
1. Base is regular octagon of side 20cm and altitude of 20cm.
2. Base is regular hexagon of side 20cm and altitude of 30cm.
3. Base is regular hexagon of side 30cm and slant height of 50cm.
these are all the same problem.
The lateral area is just the area of eight identical isosceles triangles, each with base equal to the side of the octagon, and height equal to the slant height of the pyramid, or √(h^2+a^2) where h is the height of the pyramid and a is the apothem of the base polygon.
erfe
To find the lateral area of a right pyramid, we need to calculate the sum of the areas of all the faces except the base.
1. The lateral area of a right pyramid with a regular octagon base of side 20 cm and altitude of 20 cm can be found using the formula:
Lateral Area = (Perimeter of the base) × (Slant height) / 2
The perimeter of a regular octagon can be calculated using the formula:
Perimeter = 8 × Side length
Given the side length of 20 cm, the perimeter of the base is:
Perimeter = 8 × 20 cm = 160 cm
Now, substitute the values into the lateral area formula:
Lateral Area = (160 cm) × (20 cm) / 2
Lateral Area = 3200 cm²
Therefore, the lateral area of the pyramid is 3200 cm².
2. The lateral area of a right pyramid with a regular hexagon base of side 20 cm and altitude of 30 cm can be found using the same formula:
Lateral Area = (Perimeter of the base) × (Slant height) / 2
The perimeter of a regular hexagon can be calculated using the formula:
Perimeter = 6 × Side length
Given the side length of 20 cm, the perimeter of the base is:
Perimeter = 6 × 20 cm = 120 cm
Now, substitute the values into the lateral area formula:
Lateral Area = (120 cm) × (30 cm) / 2
Lateral Area = 1800 cm²
Therefore, the lateral area of the pyramid is 1800 cm².
3. The lateral area of a right pyramid with a regular hexagon base of side 30 cm and slant height of 50 cm can be found using the same formula:
Lateral Area = (Perimeter of the base) × (Slant height) / 2
The perimeter of a regular hexagon can be calculated using the formula:
Perimeter = 6 × Side length
Given the side length of 30 cm, the perimeter of the base is:
Perimeter = 6 × 30 cm = 180 cm
Now, substitute the values into the lateral area formula:
Lateral Area = (180 cm) × (50 cm) / 2
Lateral Area = 4500 cm²
Therefore, the lateral area of the pyramid is 4500 cm².
To find the lateral area of a right pyramid, we need to calculate the sum of the areas of all the lateral faces. The formula for the lateral area of a pyramid is given by:
Lateral Area = (Perimeter of the Base) × (Slant Height) / 2
Let's calculate the lateral areas for each of the given pyramids:
1. Base is a regular octagon of side 20cm and altitude of 20cm:
To find the perimeter of the base, we need to find the length of one side of the octagon. Since it is a regular octagon, all sides are equal.
To calculate the side length, we can use the formula:
Side Length = 2 × (apothem) × tan(π/8)
Here, the apothem is given as the altitude of the pyramid, which is 20cm.
Side Length = 2 × 20cm × tan(π/8)
Side Length ≈ 29.09 cm (rounded to two decimal places)
Now that we have the side length, we can find the perimeter of the octagon:
Perimeter = 8 × Side Length
Perimeter ≈ 8 × 29.09 cm ≈ 232.72 cm (rounded to two decimal places)
Finally, we can calculate the lateral area:
Lateral Area = (Perimeter) × (Slant Height) / 2
Lateral Area ≈ 232.72 cm × 20 cm / 2 ≈ 4654.4 cm² (rounded to one decimal place)
Therefore, the lateral area of the given pyramid is approximately 4654.4 cm².
2. Base is a regular hexagon of side 20cm and altitude of 30cm:
The process is similar to the previous example.
To find the perimeter of the hexagon, we need to multiply the side length by 6 since all sides are equal:
Perimeter = 6 × Side Length
Perimeter = 6 × 20 cm = 120 cm
Now we can calculate the lateral area:
Lateral Area = (Perimeter) × (Slant Height) / 2
Lateral Area = 120 cm × 30 cm / 2 = 1800 cm²
Therefore, the lateral area of the given pyramid is 1800 cm².
3. Base is a regular hexagon of side 30cm and slant height of 50cm:
We already have the side length and the slant height.
To find the perimeter of the hexagon:
Perimeter = 6 × Side Length
Perimeter = 6 × 30 cm = 180 cm
Now we can calculate the lateral area:
Lateral Area = (Perimeter) × (Slant Height) / 2
Lateral Area = 180 cm × 50 cm / 2 = 4500 cm²
Therefore, the lateral area of the given pyramid is 4500 cm².