An illustration shows a 3 D triangular prism. The side of the base is 4.5 centimeters. The perpendicular height of the triangular base is 3.9 centimeters. The perpendicular height of the lateral face from the apex to the center of the base is 3.5 centimeters. There are two right angle symbols to show the perpendicular lines. The heights are shown as dashed lines.
A metalworker is creating decorative points in the form of triangular pyramids to go on top of a fence she is building. She wants to know the surface area of each pyramid. Use the surface area formula to find the surface area of each pyramid based on the figure.
The surface area of a triangular pyramid can be calculated using the formula:
Surface Area = 0.5 x base perimeter x slant height + base area
For the given triangular prism, the base is a triangle with a side length of 4.5 cm and a perpendicular height of 3.9 cm. The area of the base can be calculated as:
Base area = 0.5 x base x height
Base area = 0.5 x 4.5 cm x 3.9 cm
Base area = 0.5 x 4.5 cm x 3.9 cm
Base area = 8.775 cm^2
The slant height of the triangular face can be calculated as the hypotenuse of a right triangle with legs of 3.9 cm and 3.5 cm:
Slant height = √(3.9 cm)^2 + (3.5 cm)^2
Slant height = √(15.21 cm + 12.25 cm)
Slant height = √27.46 cm
Slant height ≈ 5.24 cm
Now, we can use the surface area formula to find the surface area of the triangular pyramid:
Surface Area = 0.5 x base perimeter x slant height + base area
Surface Area = 0.5 x 4.5 cm + 4.5 cm + 4.5 cm + 3.5 cm x 5.24 cm + 8.775 cm^2
Surface Area = 0.5 x 13.5 cm x 5.24 cm + 8.775 cm^2
Surface Area = 34.02 cm^2 + 8.775 cm^2
Surface Area ≈ 42.795 cm^2
Therefore, the surface area of each pyramid is approximately 42.795 square centimeters.