a hexagonal pyramid has a base with and area of 25 in2 and is 7 in tall what is the volume of the pyramid rounded to the nearest whole number
V = 1/3(AH)
A =25in^2, H = 7in
V = 1/3(25*7) = 58.3333in^3
To find the volume of a pyramid, you need to know the area of the base and the height.
In this case, you are given that the base of the hexagonal pyramid has an area of 25 square inches (in^2) and the height is 7 inches (in).
The formula for the volume of a pyramid is:
Volume = (1/3) * Area of Base * Height
Let's calculate it step by step.
Step 1: Find the area of one triangle in the hexagonal base.
A hexagon's base is composed of 6 congruent triangles. Since the area of the whole base is 25 square inches, the area of one triangle would be:
Area of one triangle = Area of Base / Number of Triangles
Area of one triangle = 25 in^2 / 6 triangles
Step 2: Calculate the height of the triangular face from the center to one of its corners.
To do this, we can imagine drawing a perpendicular line from the apex (top point) of the pyramid to the middle of the base. This line would be half the length of an apothem (distance from the center to any of the corners). Since the hexagon has equal sides, the height of this line would be equal to the height of one of the triangles in the base.
Step 3: Calculate the apothem by using the Pythagorean theorem.
The height (h) of one of the triangles is given as 7 inches. To find the length of the apothem (a), we can use the right triangle formed with the height as one side and half the length of one side of the hexagon as the other side. Since the triangle is right-angled, we can use the Pythagorean theorem:
a^2 + (s/2)^2 = h^2
a^2 + (s/2)^2 = 7^2
Solve for a.
Step 4: Calculate the area of one triangle in the base.
Now that we know the height (h) of one triangle and the apothem (a), we can calculate the area of one triangle (T):
Area of one triangle = (1/2) * base * height
Since the base of the triangle is equal to the apothem (a) and the height is given as 7 inches, we can calculate the area of one triangle.
Step 5: Calculate the volume of the pyramid.
Finally, we can use the formula to find the volume of the pyramid:
Volume = (1/3) * Area of Base * Height
Plug in the values of the area of one triangle and the given height to calculate the volume. Round the result to the nearest whole number.
Note: It's important to follow these steps and make sure to substitute the correct values into the formulas to arrive at the correct answer.
v = 1/3 Bh
plug in your numbers