if a square pyramid has a base side length of 6.3 m and a slant height of 4 m. find its surface area

Consider one of the four isosceles triangles.

drawing a line from the top vertex to the midpoint of the base gives you a right-angled triangle whose height can be found by

h^2 + 3.15^2 = 4^2
h = √6.0775

the surface area of the four triangles is then
4*(1/2)(6.3)51√6.0775
= 31.06

if you want to include the area of the base to your answer, add 6.3^2

To find the surface area of a square pyramid, you need to calculate the area of its base and the sum of the areas of its four faces.

Step 1: Calculate the area of the base.
The base of a square pyramid is a square, and its area is calculated by squaring the length of one of its sides.

Area of the base = (side length)^2
Area of the base = (6.3 m)^2
Area of the base = 39.69 m^2

Step 2: Calculate the area of the four faces.
Each face of the pyramid is a triangle with the slant height as the height and one side being the base side. So, we need to calculate the area of one triangle and then multiply it by 4 to account for all four faces.

Area of a triangle = (1/2) * base * height
Area of a triangle = (1/2) * 6.3 m * 4 m
Area of a triangle = 12.6 m^2

Area of the four faces = 4 * Area of a triangle
Area of the four faces = 4 * 12.6 m^2
Area of the four faces = 50.4 m^2

Step 3: Calculate the total surface area.
The total surface area is the sum of the area of the base and the area of the four faces.

Total surface area = Area of the base + Area of the four faces
Total surface area = 39.69 m^2 + 50.4 m^2
Total surface area = 90.09 m^2

Therefore, the surface area of the square pyramid is 90.09 square meters.

To find the surface area of a square pyramid, you need to calculate the area of each individual face and then add them together. A square pyramid has four triangular faces and one square base.

1. Start by calculating the area of the base. Since the base is square, its area is given by the formula: Area = side × side. In this case, the side length of the base is 6.3 m, so the area of the base is (6.3 m) × (6.3 m) = 39.69 m².

2. Next, calculate the area of each triangular face. The formula for the area of a triangle is: Area = 0.5 × base × height. In a square pyramid, the slant height is both the height of the triangle and the slant height of the pyramid. Since we are given a slant height of 4 m, the height of the triangle is also 4 m.

a. To find the base of each triangle, divide the side length of the base by 2. In this case, the base length is 6.3 m, so the base of each triangle is 6.3 m ÷ 2 = 3.15 m.
b. Apply the formula for the area of each triangle: Area = 0.5 × base × height = 0.5 × 3.15 m × 4 m = 6.3 m².

3. Since there are four triangular faces, the total area of all the triangular faces is 4 × 6.3 m² = 25.2 m².

4. Finally, add the area of the base and the total area of the triangular faces to find the surface area of the square pyramid: Surface Area = Base Area + Triangular Faces Area = 39.69 m² + 25.2 m² = 64.89 m².

Therefore, the surface area of the square pyramid is 64.89 m².