find the surface area of a square pyramid with s=4.2 cm and h=7cm( slant height)

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Well, well, well! Let's calculate the surface area of that square pyramid, shall we? Now, a square pyramid consists of a square base and four triangular faces.

To find the surface area, we need to find the area of the square base and the area of each triangular face.

First, let's find the area of the square base. Since each side of the base is given as 4.2 cm, we can simply square this number to find the area:

Area of the square base = (4.2 cm)^2 = 17.64 cm^2

Next, we'll tackle the triangular faces. Each face is an isosceles triangle, and we know the slant height (h) is 7 cm. But since the slant height is not the base of the triangle, we need to calculate it using a bit of good ol' Pythagoras.

Let's call the height of the triangle (the distance from the tip to the base) "t." We can find the value of t using the Pythagorean theorem:

t^2 = h^2 - (s/2)^2
t^2 = 7^2 - (4.2/2)^2
t^2 = 49 - 2.205
t^2 = 46.795
t ≈ √46.795
t ≈ 6.83 cm

Now that we have the height (t) of the triangle, we can calculate the area of each triangular face using the formula:

Area of a triangle = (1/2) * base * height

For each triangular face, the base is equal to the side length (s), and the height is equal to t. So, the area of each face is:

Area of each triangular face = (1/2) * s * t
Area of each triangular face ≈ (1/2) * 4.2 cm * 6.83 cm
Area of each triangular face ≈ 14.3837 cm^2

Since we have four triangular faces, we just need to multiply the area of one face by 4:

Total surface area = Area of square base + 4 * Area of triangular face
Total surface area ≈ 17.64 cm^2 + 4 * 14.3837 cm^2
Total surface area ≈ 17.64 cm^2 + 57.5348 cm^2
Total surface area ≈ 75.1748 cm^2

So, my friend, the surface area of that square pyramid is approximately 75.1748 cm^2.

To find the surface area of a square pyramid, you'll need to calculate the area of each face separately and then sum them up.

Let's break down the process step by step:

1. Determine the base area:
- Since the pyramid is square-based, the base area is found by squaring the length of one side (s) of the square.
- In this case, s = 4.2 cm, so the base area (A_base) is calculated as A_base = s^2.

2. Calculate the lateral area:
- The lateral area represents the sum of the areas of the four triangular faces of the pyramid.
- To find the area of a triangle, you can use the formula: A_triangle = 0.5 * base * height.
- The base of each triangular face is equal to the length of one side of the base square (s), and the height is the slant height (h).
- In this case, h = 7 cm, so the lateral area (A_lateral) is calculated as A_lateral = 4 * (0.5 * s * h).

3. Add the base area and the lateral area to get the total surface area:
- The surface area (A_total) is equal to the sum of the base area (A_base) and the lateral area (A_lateral).
- So, A_total = A_base + A_lateral.

Let's apply the formulas to find the surface area of the square pyramid:

1. Calculate the base area:
A_base = s^2 = 4.2 cm * 4.2 cm = 17.64 cm^2

2. Calculate the lateral area:
A_lateral = 4 * (0.5 * s * h) = 4 * (0.5 * 4.2 cm * 7 cm) = 58.8 cm^2

3. Calculate the total surface area:
A_total = A_base + A_lateral = 17.64 cm^2 + 58.8 cm^2 = 76.44 cm^2

Therefore, the surface area of the square pyramid is 76.44 cm^2.

the surface is 4 triangles with

base = 4.2
height = 7

lateral area is thus 4 * 1/2 (4.2)(7)

add the square base if desired.