A regular triangular pyramid has slant height 6 cm and base edges of length 15 cm. Find the volume of the pyramid and the surface area.
To find the volume and surface area of a regular triangular pyramid, we can use the following formulas:
1. Volume of a pyramid: V = (1/3) * base area * height
2. Surface area of a pyramid: A = base area + (1/2) * perimeter of base * slant height
Let's calculate the volume first.
The base of the pyramid is a regular triangle, so we can use the formula for the area of an equilateral triangle:
Area of an equilateral triangle: A = (sqrt(3)/4) * side^2
In this case, the side length of the equilateral triangle (base of the pyramid) is 15 cm. So,
Base area = (sqrt(3)/4) * (15^2)
Next, we need to find the height of the pyramid. We can use the Pythagorean Theorem on one of the triangular faces of the pyramid.
The height is the perpendicular distance from the vertex to the base of the triangle. Consider a right triangle formed by the height, half of one of the base edges (7.5 cm), and the slant height (6 cm).
We can use the Pythagorean Theorem to solve for the height:
height^2 = slant height^2 - (half base edge)^2
= 6^2 - 7.5^2
Once we have the height, we can calculate the volume using the formula V = (1/3) * base area * height.
Now let's move on to calculating the surface area.
The base area is the same as what we calculated for the volume.
The perimeter of the base is simply 3 times the length of one of the base edges, since it is a regular triangle.
The surface area formula is A = base area + (1/2) * perimeter of base * slant height.
Now let's plug in the values and calculate the volume and surface area.