How to find the surface area of a hexagonal pyramid?
Understand the Problem
The question is asking for the method to calculate the surface area of a hexagonal pyramid. To solve it, we will use the formula for the surface area that includes both the base area and the lateral area of the pyramid.
Answer
The surface area of a hexagonal pyramid is given by: $$ A_{total} = \frac{3\sqrt{3}}{2} s^2 + 3 s h $$
Answer for screen readers
The formula for the surface area of a hexagonal pyramid is: $$ A_{total} = \frac{3\sqrt{3}}{2} s^2 + 3 s h $$
Steps to Solve
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Identify the base area To find the surface area of a hexagonal pyramid, we first need to calculate the area of the hexagonal base. The formula for the area of a regular hexagon with side length $s$ is given by: $$ A_{base} = \frac{3\sqrt{3}}{2} s^2 $$
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Determine the lateral area Next, we calculate the lateral area of the pyramid. The lateral area consists of 6 triangular faces. Each triangle has a base of length $s$ and height $h$, where $h$ is the slant height of the pyramid. The area of one triangular face is: $$ A_{triangle} = \frac{1}{2} \times s \times h $$
Thus, the lateral area for 6 triangles is: $$ A_{lateral} = 6 \times A_{triangle} = 6 \times \frac{1}{2} s h = 3s h $$
- Calculate the total surface area Finally, add the base area and the lateral area together to find the total surface area of the hexagonal pyramid: $$ A_{total} = A_{base} + A_{lateral} $$ Substituting the previous expressions gives: $$ A_{total} = \frac{3\sqrt{3}}{2} s^2 + 3 s h $$
The formula for the surface area of a hexagonal pyramid is: $$ A_{total} = \frac{3\sqrt{3}}{2} s^2 + 3 s h $$
More Information
This formula allows you to compute the surface area of a hexagonal pyramid, which includes both the area of its hexagonal base and the area of its triangular lateral faces. The presence of the square root in the formula for the base reflects the geometric properties of hexagons.
Tips
- Forgetting to calculate both the base area and the lateral area separately before adding them together.
- Miscalculating the area of the triangular faces, especially by mixing up base and height.
- Not using the correct dimensions for the slant height when determining the lateral area.
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