Use the image to answer the question.
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 = Area of the base + Area of the three triangular faces.
From the given information, we know that the base of the pyramid is a triangle with a side length of 4.5 centimeters and a perpendicular height of 3.9 centimeters. So, the area of the base can be calculated as:
Area of the base = (1/2) * base * height
= (1/2) * 4.5 cm * 3.9 cm
= 8.775 cm^2
Now, let's calculate the areas of the three triangular faces. Each face is a right-angled triangle, with one side equal to the height of the lateral face, which is 3.5 centimeters, and the other two sides equal to the slant height of the pyramid, which is the distance from the apex to the center of the base.
To calculate the slant height, we can use the Pythagorean theorem. The slant height (l) is the hypotenuse, and the base of the right-angled triangle is half the length of the base of the pyramid, which is (1/2) * 4.5 centimeters = 2.25 centimeters. The height of the right-angled triangle is given as 3.5 centimeters.
Using the Pythagorean theorem:
l^2 = (2.25 cm)^2 + (3.5 cm)^2
= 5.0625 cm^2 + 12.25 cm^2
= 17.3125 cm^2
l ≈ √(17.3125 cm^2) ≈ 4.16 cm (rounded to two decimal places)
Now, let's calculate the areas of the three triangular faces:
Area of each triangular face = (1/2) * base * height
= (1/2) * 2.25 cm * 3.5 cm
= 3.9375 cm^2
Since there are three triangular faces, the total surface area of the pyramid can be calculated as:
Surface Area = Area of the base + 3 * Area of each triangular face
= 8.775 cm^2 + 3 * 3.9375 cm^2
= 25.5875 cm^2
Therefore, the surface area of each triangular pyramid is approximately 25.59 square centimeters.