Name: Volume of 3D Shapes
Uploaded: 2019-07-19
Duration: 7 min 3 s
Description: Here's how to calculate the three-dimensional volume that the shape will occupy. This is like the amount of water that a shape could hold if it was hollow.

Question

$A B C$ is the cross-section of the prism. Find the surface area and volume of the prism if $A B = A C = 5 c m$ , $B C = 6 c m$ and $C R = 8 c m$

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Accepted Answer

We have,

$A B C$ is the cross-section of the prism. If $A B = A C = 5 c m$ , $B C = 6 c m$ and $C R = 8 c m$

Now we know that,

The area of a triangle is $\sqrt{s (s - a) (s - b) (s - c)}$ , where $s$ is the semi-perimeter, $a, b, c$ are the sides of a triangle

Then, in $∆ A B C$

$s = \frac{5 + 5 + 6}{2} = 8 c m$

Then, area of the $∆ A B C = \sqrt{8 (8 - 5) (8 - 5) (8 - 6)}$

$= \sqrt{8 \times 3 \times 3 \times 2} = 12 c m^{2}$

Now area of $∆ A B C =$ Area of $∆ P Q R$

Then, area of $∆ A B C +$ area of $∆ P Q R = 12 + 12 = 24 c m^{2}$

Now the area of a rectangle is $(length) (breadth)$

Then, area of the face $A C R P = (5) (8) = 40 c m^{2}$

Area of the face $A C R P =$ Area of the face $A B Q P$

Then, area of face $A C R P +$ area of face $A B Q P = 40 + 40 = 80 c m^{2}$

And the area of the face $B C R Q = (6) (8) = 48 c m^{2}$

The surface area of the prism $=$ Area of $∆ A B C +$ Area of $∆ P Q R +$ Area of the face $A C R P +$ Area of the face $A B Q P +$ Area of the face $B C R Q$

Therefore, the surface area of the prism $= 12 + 12 + 40 + 40 + 48 = 152 c m^{2}$

And the volume of the prism $=$ (Surface area of the cross-section $∆ A B C$ ) $\times$ (Length)

$= 12 \times 8 = 96 c m^{3}$

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