Question

The ratio of the heights of two similar prisms is \(5: 3\). a. Find the ratio of their surface areas. b. Find the ratio of their volumes.

Short answer

a. The ratio of surface areas is \(25:9\). b. The ratio of volumes is \(125:27\).

Step-by-step solution

Understanding the Ratio of Heights

We are given that the ratio of the heights of two similar prisms is \(5:3\). This will be our starting point for calculating the ratios of surface areas and volumes since the dimensions of similar prisms are proportional.

Ratio of Surface Areas

For similar figures, the ratio of their surface areas is the square of the ratio of their linear dimensions (heights, in this case). So, the ratio of the surface areas is calculated as \((5:3)^2 = \frac{5^2}{3^2} = \frac{25}{9}\).

Ratio of Volumes

For similar figures, the ratio of their volumes is the cube of the ratio of their linear dimensions. So, the ratio of the volumes is \((5:3)^3 = \frac{5^3}{3^3} = \frac{125}{27}\).

Key concepts

Similar Prisms

Similar prisms are geometric shapes that have the same shape but may differ in size. This means that each dimension, such as length, width, and height, of one prism, is proportional to the corresponding dimension of the other. When two prisms are similar, their corresponding angles are identical, and their side lengths are proportional.

• Imagine two towers made from the same blueprint but at different scales. Each aspect of the smaller tower is just a scaled-down version of the larger one.

Understanding similar prisms is crucial because it allows us to extrapolate various properties and measurements, such as surface area and volume, based on the proportionality of one dimension, as we will see in the sections that follow.

Ratio of Surface Areas

The surface area of a 3-dimensional object is the total area that the surface of the object covers. For similar prisms, the ratio of their surface areas is not simply the same as the ratio of their linear dimensions, such as heights. Instead, the ratio of the surface areas is found by squaring the ratio of the linear dimensions.To visualize this, consider that if the corresponding sides of prisms have a ratio of 5:3, then the surface areas will have a ratio of \[ \left( \frac{5}{3} \right)^2 = \frac{25}{9} \].

• If one prism's side is 5 units long and the other's is 3 units long, the bigger prism doesn't just have sides 5/3 times longer, but has a more significant total surface area coverage by a factor of 25/9.

This concept of squaring the linear proportion extends from the fact that surface area involves two dimensions.

Ratio of Volumes

When dealing with volumes of similar prisms, the calculation changes from surface area because volume is a three-dimensional property. For similar prisms, the ratio of their volumes is the cube of the ratio of their linear dimensions.To understand why, think of volume as how much space a figure occupies. If the dimensions of one prism are 5 times a unit measure and the other is 3, then the volume will scale according to \[ \left( \frac{5}{3} \right)^3 = \frac{125}{27} \].

• This makes sense because volume considers three dimensions: length, width, and height, which leads us to factor the proportionality three times.

So, when comparing the volumes of similar prisms, remember to cube the linear dimension ratio. This provides a clear picture of how the space inside the prism scales with size.