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### 1. Total Area of a Regular Pyramid with a Rhombus Base
Given:
- Height of the pyramid (\( h \)) = 16 cm
- Side length of the rhombus base = 24 cm
- One diagonal of the rhombus = 10 cm

To find the total area, we need:
1. **Area of the base** (the rhombus).
2. **Area of the four triangular faces** of the pyramid.

#### Step 1: Area of the Rhombus Base
The area \( A \) of a rhombus with diagonals \( d_1 \) and \( d_2 \) is:
\[
A = \frac{1}{2} \times d_1 \times d_2
\]
Here, \( d_1 = 10 \) cm. We can find \( d_2 \) using the Pythagorean theorem, since each side of the rhombus is 24 cm.

Let's call the half-diagonal \( d_1/2 = 5 \) cm. Since we know the length of each side of the rhombus (24 cm), we can find the other half-diagonal (let’s call it \( x \)):

\[
24^2 = 5^2 + x^2
\]
\[
576 = 25 + x^2
\]
\[
x^2 = 551 \Rightarrow x = \sqrt{551} \approx 23.45 \text{ cm}
\]

Thus, the other diagonal \( d_2 = 2 \times 23.45 \approx 46.9 \) cm.

Now, the area of the rhombus is:
\[
A = \frac{1}{2} \times 10 \times 46.9 \approx 234.5 \, \text{cm}^2
\]

#### Step 2: Area of the Triangular Faces
Each triangular face has a slant height \( l \), which we can find using the Pythagorean theorem. Since the height of the pyramid is 16 cm, we need to find the slant height from the apex to the midpoint of a side of the rhombus.

To proceed with the calculations, I’ll calculate each face's area assuming each triangular face and sum up all areas for the total area.

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### 2. Total Area of a Regular Tetrahedron with Each Side 6



1. Find the total area of a regular pyramid 16 cm high, with a rhombus as a base with 24 cm on its side and 10 cm in one diagonal.
2. Find the total area of a regular tetrahedron if each side is 6.

Learn how to find the total surface area of a regular pyramid with a rhombus base and a height of 16 cm, as well as the total area of a regular tetrahedron with each side measuring 6 cm. This problem involves geometry concepts, including area calculations for rhombuses, pyramids, and tetrahedrons, with step-by-step solutions.

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