Final answer:
To find the surface area of the pyramid: 1. Calculate the area of the base by squaring the length of one side. 2. Find the area of each triangular face using the altitude and the base edge. 3. Add the area of the base and the area of the triangular faces to get the surface area of the pyramid.
Step-by-step explanation:
To find the surface area of the pyramid, we need to calculate the areas of all the faces and add them together. Let's start by finding the area of the base. Since it's a square, we can use the formula for the area of a square: A = s^2, where s is the length of a side. In this case, if the base edge is 7 cm long, the area of the base is 7^2 = 49 cm^2.
Next, let's find the area of each triangular face. Since the pyramid is a regular square pyramid, all the triangular faces are congruent. To find the area of each triangular face, we can use the formula A = 1/2 * base * height. In this case, the base is the base edge of the triangular face, which is 7 cm. The height is the length of the altitude, which we can find using trigonometry.
Given that the angle formed by the altitude and the base edge is 20°, the ratio of the opposite side (the altitude) to the adjacent side (the base edge) is equal to the tangent of the angle. So, we can use the formula: tan(20°) = opposite/adjacent. Rearranging the formula, we get: opposite = tan(20°) * adjacent. Plugging in the values, opposite = tan(20°) * 7 cm = 2.41 cm.
Now that we know the length of the altitude, we can find the area of each triangular face: A = 1/2 * 7 cm * 2.41 cm = 8.435 cm^2. Since there are 4 of these faces, the total area of the triangular faces is 4 * 8.435 cm^2 = 33.74 cm^2.
Finally, we can find the surface area of the pyramid by adding the area of the base and the area of the triangular faces: Surface Area = Base Area + Triangular Faces Area = 49 cm^2 + 33.74 cm^2 = 82.74 cm^2. Therefore, the surface area of the pyramid is approximately 82.74 cm^2.