Volume of 3D Shapes

To find the volume we simply have to multiply all three lengths together:

 

\text{volume}= 3\times12 \times16=576\text{cm}^3

The volume of a square base pyramid is given by the following formula:

 

\text{volume}= \dfrac{1}{3} \times \text{base area} \times \text{height}

 

Substituting the values given in the question into the equation above, we find that,

 

\text{volume}= \dfrac{1}{3} \times 5^{2} \times 12 =100 \text{m}^3

So, to work out the volume of a prism we must multiply the area of the cross section by the length. In this case, the cross section is a trapezium, and the area of the trapezium is:

 

\text{area of cross section }=\dfrac{1}{2}\times(45+60)\times20= 1,050\text{ cm}^2

 

The length of the prism is 80 cm, so we get:

 

\text{volume of prism }=1,050\times80=84,000\text{ cm}^3

This may seem a little different, but it actually comes up a lot.

First, we must write the volume in terms of x. The volume of a pyramid is one third of the area of the base times by the perpendicular height. We know the area of the base is 18\text{cm}^2, and the expression we’re given for the height is x+5, so the volume is

 

\dfrac{1}{3}\times18\times(x+5)=6(x+5)

 

Now, the question also gave us the volume: 54\text{cm}^3, so we can equate this value to the expression we found above and voila, we have an equation:

 

6(x+5)=54

 

Now, we solve this equation to find x. First, divide both sides by 6 to get:

 

x+5=54\div6=9

 

Then, subtracting 5 from both sides we get the answer to be:

 

x=9-5=4\text{cm}

We will work out the volume of the cylinder first, then the hemisphere, and add together the values.

 

A cylinder is a prism which means that to find the volume, we must multiply the area of the circular cross section by its length. The radius of the circle is 2.3m and the length is 5.6m, so we get:

 

\text{volume of cylinder }=\pi \times (2.3)^2\times5.6 \approx 93.07 \text{m}^3

 

Note: Keep the full answer stored in the calculator for adding it to the other value at the end.

 

Next, we are given the formula for the volume of a sphere, so to find the volume of the hemisphere we will use this formula and then half the result. The radius of the hemisphere is the same as that of the cylinder, 2.3, so we get:

 

\text{volume of hemisphere }=\dfrac{1}{2}\times\left(\dfrac{4}{3}\pi\times(2.3)^3\right)\approx 25.48\text{m}^3

 

Therefore, the total volume of the shape is:

 

93.0665...+25.4825...=119\text{ m}^3 \text{ (3s.f.)}