Question

Use the net to find the surface area of the regular pyramid.

A net includes four equilateral triangles. One triangle is shaded and it is a base. Each unshaded triangle shares a side with the shaded triangle. The shaded triangle has a side edge of 10 millimeters and area of base is 43.3 squared millimeters. The height of each unshaded triangle is 9 millimeters.


mm2

Answer 1

Answer: The surface area is 173.2 mm²

Explanation: If the four triangles are equilateral and one of the sides is 10 then all the other sides must be 10 as well, meaning that the surface area of each triangle is 43.3 mm². That means that to find the surface area of the whole pyramid you must multiply 43.3 by 4 because there are 4 separate triangles.

43.3 × 4 = 173.2

Answer 2

The surface area of the regular pyramid is 178.3 square millimeters.

To calculate the surface area of the regular pyramid, we first need to determine the area of one of the unshaded triangles (the lateral faces of the pyramid) and then use this information to find the total surface area, which includes the base and all the lateral faces.

The area of each unshaded triangle (lateral face) can be calculated using the formula for the area of a triangle:
`\[ \text{Area} = (1)/(2) * \text{base} * \text{height} \]`

Given that the base of each lateral face (which is also a side of the base triangle) is 10 millimeters, and the height (the perpendicular height from the base to the apex of each triangle) is 9 millimeters, the area of one unshaded triangle would be:
`\[ \text{Area of one unshaded triangle} = (1)/(2) * 10 * 9 = 45.0 \text{ mm}^2 \]`

Since there are three unshaded triangles, the total area of these triangles is three times the area of one triangle:
`\[ \text{Total area of unshaded triangles} = 3 * 45.0 = 135.0 \text{ mm}^2 \]`

Now, to find the surface area of the pyramid, we add the total area of the unshaded triangles to the area of the base. We are given that the area of the base is 43.3 square millimeters.

Therefore, the total surface area is:
`\[ \text{Surface area} = \text{Base area} + \text{Total area of unshaded triangles} \] \[ \text{Surface area} = 43.3 + 135.0 = 178.3 \text{ mm}^2 \]`

In conclusion, the surface area of the regular pyramid is 178.3 square millimeters.