A pyramid with vertical height of 1ocm stands on a square base of sides 20cm .
a. Calculate to 3 significant figure for its total surface area.
b. If the pyramid is hollow what it's capacity in litres.
You don't say if in your surface ares you want to include the square base.
area of the 4 triangular sides:
need the height of one of them, 10^2 + 10^2= h^2
h = 10√2 cm
Area of one triangle = (1/2)(base)(height)
= (1/2)(20)10√2 = 100√2
add the area of the base which would be 400, if it should be included, you decide
vol = (1/3)(area of base)(height) <---- notice this height is the one given
= ....
I'm not sure what "1o" centimeters means. But here's the formula for the surface area of a pyramid:
B + (1/2)(ps)
where B=area of the base, p=perimeter of the base and s=slant height. Use the Pythagorean Theorem to find the slant height.
For part b, you'll need the formula for the volume of a pyramid. Here it is:
(1/3)Bh
where B=area of the base and h=vertical height. That's all the information you need to solve this yourself.
a. To calculate the total surface area of the pyramid, we need to calculate the area of each face and then sum them up.
1. Base area:
The base of the pyramid is a square, so we can calculate its area by multiplying its side length by itself.
Area of the base = side length * side length = 20 cm * 20 cm = 400 cm^2
2. Lateral faces:
The pyramid has four triangular lateral faces, each with the same dimensions. To calculate the area of one triangular face, we need to find the height of the triangle.
Using the Pythagorean theorem, we can find the height:
Height^2 = (Base/2)^2 + Height^2
Height^2 = 10^2 + Height^2
2 * Height^2 = 100
Height^2 = 50
Height = √50 cm ≈ 7.071 cm
Now, we can calculate the area of one triangular face:
Area of one triangular face = (Base * Height) / 2 = (20 cm * 7.071 cm) / 2 ≈ 70.71 cm^2
3. Total surface area:
The total surface area is the sum of the base area and the areas of the four triangular faces:
Total surface area = Base area + 4 * Area of one triangular face
Total surface area = 400 cm^2 + 4 * 70.71 cm^2 ≈ 709.84 cm^2
Therefore, the total surface area of the pyramid is approximately 709.84 cm^2 (to 3 significant figures).
b. To find the capacity of the hollow pyramid in liters, we need to calculate the volume of the hollow space.
1. Volume of the pyramid:
The volume of a pyramid can be calculated by multiplying the area of the base by the height and then dividing by 3.
Volume of the pyramid = (Base area * Height) / 3
Volume of the pyramid = (400 cm^2 * 10 cm) / 3 ≈ 1333.33 cm^3
2. Volume of the solid pyramid:
To calculate the volume of the solid pyramid, we need to subtract the volume of the hollow space. Since the hollow space is not clearly defined in the question, we will assume that it forms another pyramid with a smaller base.
The volume of the solid pyramid is equal to the volume of the larger pyramid minus the volume of the smaller pyramid.
3. Calculate the volume of the hollow space:
To determine the volume of the smaller pyramid, we can use similar triangles. Since the dimensions of the smaller pyramid are not provided, we need to make an assumption. Let's assume that the smaller pyramid has a height of 5 cm.
Using similar triangles, we can set up the following ratio:
(Base of the smaller pyramid) / (Base of the larger pyramid) = (Height of the smaller pyramid) / (Height of the larger pyramid)
Let x be the Base of the smaller pyramid:
x / 20 cm = 5 cm / 10 cm
Simplifying the equation:
x = (5 cm / 10 cm) * 20 cm = 10 cm
Now that we know the base and height of the smaller pyramid, we can calculate its volume using the same formula:
Volume of the smaller pyramid = (Base area * Height) / 3
Volume of the smaller pyramid = (100 cm^2 * 5 cm) / 3 ≈ 166.67 cm^3
4. Calculate the volume of the hollow space:
Volume of the hollow space = Volume of the larger pyramid - Volume of the smaller pyramid
Volume of the hollow space = 1333.33 cm^3 - 166.67 cm^3 = 1166.67 cm^3
5. Convert the volume to liters:
Since 1 liter is equal to 1000 cm^3, we can convert the volume of the hollow space.
Volume of the hollow space in liters = 1166.67 cm^3 / 1000 ≈ 1.17 liters
Therefore, if the pyramid is hollow, its capacity is approximately 1.17 liters.