Final answer:
To find the total surface area of the square-based pyramid, we can calculate the area of the base and the area of the four triangular faces. By using the given length of the base, we can determine the area of the base and the height of the pyramid. Finally, we can calculate the area of one triangular face and multiply it by four to find the total surface area of the pyramid.
Step-by-step explanation:
To find the total surface area of a square-based pyramid, we need to calculate the area of the base and the area of the four triangular faces. The length of the base is given as 18 cm, so the area of the base is 18 cm × 18 cm = 324 cm². Since a square-based pyramid has four triangular faces that are congruent, we can find the area of one triangular face and multiply it by four to get the total. Let's say the height of the pyramid is 'h'. The area of a triangle is given by the formula 1/2 × base × height. In this case, the base of the triangle is 18 cm (same as the base length of the square) and the height is 'h'. So, the area of one triangular face is 1/2 × 18 cm × 'h'. To find 'h', we can use the total surface area of the pyramid, which is given as 1,296 cm². The total surface area is the sum of the area of the base and the area of the four triangular faces. So, 1,296 cm² = 324 cm² + 4 × (1/2 × 18 cm × 'h'). Simplifying this equation will give us 'h'.
Let's solve for 'h':
1,296 cm² = 324 cm² + 4 × (1/2 × 18 cm × 'h')
1,296 cm² = 324 cm² + 4 × 9 cm × 'h'
1,296 cm² = 324 cm² + 36 cm × 'h'
972 cm² = 36 cm × 'h'
'h' = 972 cm² / 36 cm
'h' = 27 cm
Now that we have the height of the pyramid, we can calculate the area of one triangular face and multiply it by four to find the total surface area:
Area of one triangular face = 1/2 × base × height
= 1/2 × 18 cm × 27 cm
= 243 cm²
Total surface area = 324 cm² + 4 × 243 cm²
= 972 cm²
Therefore, the total surface area of the pyramid is 972 cm².