A rhombic prism has the diagonal of the rhombus measuring 16 and 12 cm. If the the altitude of the prism is equal in size to the side of the rhombus. What is the surface area and the volume of the Rhombic Prism?

Question

A rhombic prism has the diagonal of the rhombus measuring 16 and 12 cm.  If the the altitude of the prism is equal in size to the side of the rhombus. What is the surface area and the volume of the Rhombic Prism?

Avery Allen · Accepted Answer

Total Surface Area of Rhombuic Prism #T S A = color(red)(592 cm^2)#

Volume of Rhobic Prism #V_R = color (blue)(960 cm^3)#

Area of Rhombus base #A_R = (1/2) * D1 * D2 = (1/2) * 16 * 12 = 96 cm^2#

Side of Rhombus #a = sqrt(((D1)/2)^2 + ((D2)/2)^2)#

#a = sqrt(8^2 + 6^2) = 10 cm#

Savannah Aguilar · Answer

undefined To find the surface area and volume of the rhombic prism, we first need to calculate the side length of the rhombus.

Given that the diagonal of the rhombus measures 16 cm and 12 cm, and the altitude of the prism is equal to the side of the rhombus, we can use the Pythagorean theorem to find the side length of the rhombus.

Let $d_1 = 16$ cm and $d_2 = 12$ cm be the diagonals of the rhombus. Let $s$ be the side length of the rhombus.

By the Pythagorean theorem, we have:

$$
s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2
$$

$$
s^2 = \left(\frac{16}{2}\right)^2 + \left(\frac{12}{2}\right)^2
$$

$$
s^2 = 8^2 + 6^2
$$

$$
s^2 = 64 + 36
$$

$$
s^2 = 100
$$

$$
s = \sqrt{100}
$$

$$
s = 10
$$

So, the side length of the rhombus is 10 cm.

Now, let's find the surface area and volume of the rhombic prism.

The surface area of the rhombic prism is given by the formula:

$$
\text{Surface Area} = 4 \times \text{Area of the rhombus} + 4 \times \text{Area of the rectangle}
$$

The volume of the rhombic prism is given by the formula:

$$
\text{Volume} = \text{Area of the rhombus} \times \text{Height of the prism}
$$

Given that the altitude of the prism is equal to the side of the rhombus, the height of the prism is also 10 cm.

Now, we need to find the area of the rhombus. Since the side length is 10 cm, and the altitude is also 10 cm, the area of the rhombus is $10 \times 10 = 100 \text{ cm}^2$.

The area of the rectangle is given by the formula $s \times d$, where $s$ is the side length of the rhombus and $d$ is the distance between two opposite sides of the rhombus.

Given that the diagonals of the rhombus are 16 cm and 12 cm, the distance between two opposite sides is 12 cm. Therefore, the area of the rectangle is $10 \times 12 = 120 \text{ cm}^2$.

Substituting these values into the formulas:

Surface Area $= 4 \times 100 + 4 \times 120 = 400 + 480 = 880 \text{ cm}^2$

Volume $= 100 \times 10 = 1000 \text{ cm}^3$

Therefore, the surface area of the rhombic prism is $880 \text{ cm}^2$ and the volume is $1000 \text{ cm}^3$.