To calculate the total surface area of the triangular pyramid, we need to find the area of all its faces, which include the triangular base and the three triangular lateral faces.

Step 1: Surface Area of the Base

The base area is given as $24 \, \text{cm}^2$.

Step 2: Surface Area of the Lateral Faces

The pyramid has three identical triangular lateral faces. Each of these triangles has a base equal to the side of the equilateral triangular base and a height that we need to calculate.

Height of Each Lateral Face

The height of each lateral triangular face can be found using the Pythagorean theorem. First, we need to determine the height of the equilateral triangular base, which is the altitude of the base triangle.

For an equilateral triangle with side $s = 6 \, \text{cm}$, the height $h_b$ of the base triangle is given by: $h_b = \frac{\sqrt{3}}{2} \times s = \frac{\sqrt{3}}{2} \times 6 \, \text{cm} = 3\sqrt{3} \, \text{cm}$

Now, consider the triangular lateral face. This triangle has a base of $6 \, \text{cm}$ (same as the side of the equilateral triangle) and a height $h_l$ that can be calculated using the relationship between the slant height and the height of the pyramid.

The height of the triangular face $h_l$ is related to the slant height $s_l$ of the pyramid by the Pythagorean theorem: $s_l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}$ where $h = 10 \, \text{cm}$ is the height of the pyramid, and $\frac{s}{2} = 3 \, \text{cm}$ is half the side of the base.

Calculate the slant height $s_l$: $s_l = \sqrt{10^2 + 3^2} = \sqrt{100 + 9} = \sqrt{109} \, \text{cm}$

The area $A_l$ of each lateral face is then: $A_l = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 6 \, \text{cm} \times \sqrt{109} \, \text{cm} = 3\sqrt{109} \, \text{cm}^2$

Step 3: Total Surface Area

The total surface area $A_{total}$ of the pyramid is the sum of the area of the base and the areas of the three lateral faces: $A_{total} = A_{base} + 3 \times A_l = 24 \, \text{cm}^2 + 3 \times 3\sqrt{109} \, \text{cm}^2 = 24 \, \text{cm}^2 + 9\sqrt{109} \, \text{cm}^2$ Thus, the total surface area is: $A_{total} \approx 24 + 9 \times 10.44 \, \text{cm}^2 = 24 + 93.96 \, \text{cm}^2 = 117.96 \, \text{cm}^2$

Final Answer

The total surface area of the triangular pyramid is approximately $117.96 \, \text{cm}^2$.

Would you like more details or have any questions? Here are 5 related questions to explore:

1. How does the total surface area change if the height of the pyramid is doubled?
2. What is the volume of the triangular pyramid?
3. How do you calculate the slant height if only the total surface area is known?
4. How would the surface area change if the base were a different shape, like a square?
5. What is the relationship between the side length and the height in an equilateral triangle?

Tip: Always verify that all units are consistent when solving geometry problems to avoid errors in your calculations.