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Example 9

Integrals - Step-by-Step Mathematics

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MATHEMATICS EXPLAINED

Integrals - Example 9

8 m 3 m Base area = 36 m²

Step 1 - Identify the two shapes

The church consists of two 3D shapes:

A hexagonal prism forming the main building and a hexagonal pyramid forming the roof.

Both shapes have the same base area:

\[ B=36\text{ m}^2 \]

The prism is 8 m high and the pyramid is 3 m high.

Prism + pyramid

Step 2 - Calculate the prism's volume

The volume of a prism is the base area multiplied by its height.

\[ V=B\times h \] \[ V=36\times8 \] \[ V=288\text{ m}^3 \]

This is the volume of the main part of the church, without the roof.

Prism volume = 288 m3

Step 3 - Calculate the roof's volume

The roof is a hexagonal pyramid.

Use the formula for the volume of a pyramid.

\[ V=\frac13\times B\times h \] \[ V=\frac13\times36\times3 \] \[ V=36\text{ m}^3 \]

The pyramid is one-third of a prism with the same base area and height.

Roof volume = 36 m3

Step 4 - Calculate the total volume

Add the volume of the prism and the volume of the pyramid.

\[ V_{\text{total}} =288+36 \] \[ V_{\text{total}} =324\text{ m}^3 \]

The answer is expressed in cubic metres, because we are calculating a volume.

Total volume = 324 m3

Final answers

Volume of the prism 288 m3
Volume of the pyramid 36 m3
Total volume of the church 324 m3