# 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?

Mar 8, 2018

A Rhombus is a flat shape having $4$ equal straight sides of length $a$. The two diagonals $p \mathmr{and} q$ of a rhombus are perpendicular to each other and these bisect each other. Its opposite sides are parallel, and opposite angles are equal (it is a Parallelogram).

Area of a rhombus $K = \frac{p \cdot q}{2}$ .........(1)
And in every rhombus

$4 {a}^{2} = {p}^{2} + {q}^{2}$ .........(2)

In the given question, from (2)

$4 {a}^{2} = {16}^{2} + {12}^{2}$
$\implies 4 {a}^{2} = 256 + 144$
$\implies a = \sqrt{\frac{256 + 144}{4}}$
$\implies a = 10 \setminus c m$

Rhombus is special case of square. As such a Rhombic Prism is a special case of square prism.
1. Surface area of the Rhombic Prism.
From (1)
Area of rhombic side $K = \frac{16 \cdot 12}{2} = 96 \setminus c {m}^{2}$
Area of square side $A = {10}^{2} = 100 \setminus c {m}^{2}$
Total surface area $S = 2 \times K + 4 \times A$
$\implies S = 2 \times 96 + 4 \times 100$
$\implies S = 592 \setminus c {m}^{2}$
2. Volume of the Rhombic Prism
$V = \text{Area of rhombic face"xx"Altitude}$
$\implies V = 100 \times 10 = 1000 \setminus c {m}^{3}$