Find the lateral and surface area of the regular square pyramid. Give exact answers.
I don't understand this. I need help.
No problem! I can help you understand how to find the lateral and surface area of a regular square pyramid.
A regular square pyramid is a pyramid where the base is a square and the height of each triangular face is the same. To find the lateral area, we need to find the area of all the triangular faces.
To find the lateral area (LA), use the formula:
LA = (1/2) × Perimeter of the base × Slant height
First, you'll need to find the perimeter of the base. Since the base of a regular square pyramid is a square, you just need to multiply one side length by 4.
If the length of one side of the base is 's', then the perimeter of the base would be P = 4s.
Next, you'll need to find the slant height. The slant height (l) is the height of each triangular face, and it can be found using the Pythagorean theorem. If 'h' is the height of the pyramid and 's' is the side length of the base, then:
l^2 = h^2 + (s/2)^2
Once you find 'l', you can plug it into the lateral area formula.
For the surface area (SA), you'll need to find the lateral area as we discussed above, and also add the area of the base square.
To find the area of the base square, use the formula:
Base Area = s^2
Then, the surface area is given by:
SA = LA + Base Area
By using these formulas, you'll be able to find the lateral and surface area of a regular square pyramid with exact answers.
No problem! I'm here to help. Let's break it down step-by-step.
Step 1: Understand the parts of a regular square pyramid.
A regular square pyramid has a square base and four congruent triangular faces that meet at a single point called the apex.
Step 2: Find the lateral area of a regular square pyramid.
The lateral area is the sum of the areas of all the triangular faces. In a regular square pyramid, all the triangular faces are congruent.
To find the lateral area, we need to determine the area of one of the triangular faces and then multiply it by the number of triangular faces.
Step 2.1: Find the area of one triangular face.
The area of a triangle can be found using the formula: Area = 1/2 * base * height.
In the case of a regular square pyramid, the base of the triangular face is one side of the square base of the pyramid, and the height is the slant height (the distance from the apex to a vertex of the base).
Step 3: Find the slant height of the regular square pyramid.
To find the slant height, we need to use the base edge length (let's call it "s") and the height of the pyramid (let's call it "h").
In a regular square pyramid, the slant height is the hypotenuse of a right triangle, with one leg being the height of the pyramid and the other leg being half of one side of the square base.
Using the Pythagorean theorem, we can find the slant height:
s^2 = (1/2s)^2 + h^2
s^2 = 1/4s^2 + h^2
3/4s^2 = h^2
s^2 = 4/3h^2
s = √(4/3h^2)
Step 4: Calculate the area of one triangular face.
Now that we have the slant height, we can calculate the area of one triangular face using the formula from Step 2.1: Area = 1/2 * base * height.
The base of the triangle is one side of the square base, which is "s", and the height is the slant height, which we just found to be √(4/3h^2).
So, the area of one triangular face is:
Area = 1/2 * s * √(4/3h^2)
Step 5: Calculate the lateral area
To find the lateral area, we need to multiply the area of one triangular face by the number of triangular faces, which is 4 in the case of a regular square pyramid.
Lateral Area = 4 * Area of one triangular face
Step 6: Find the surface area of the regular square pyramid
The surface area of a regular square pyramid includes both the lateral area and the base area.
Since the base of the pyramid is a square, we can find its area by squaring the base edge length.
Once we have the base area and the lateral area, we can add them together to get the surface area.
Surface Area = Lateral Area + Base Area
I hope this helps! If you have any further questions, feel free to ask.