How do you find the volume of a similar shape with a scale factor of 1.5?

To find the volume of a similar shape with a scale factor of 1.5, multiply the original volume by 1.5³.

When dealing with similar shapes, the scale factor affects their dimensions proportionally. For example, if you have a shape and you scale it by a factor of 1.5, each linear dimension (length, width, height) of the shape is multiplied by 1.5. However, volume is a three-dimensional measure, so the effect of the scale factor on volume is more significant.

To understand why, consider that volume is calculated by multiplying three dimensions together (e.g., length × width × height). If each dimension is scaled by 1.5, the new volume is found by multiplying the original volume by 1.5 for each dimension. Mathematically, this is expressed as multiplying the original volume by 1.5³ (1.5 × 1.5 × 1.5), which equals 3.375.

So, if you know the volume of the original shape, you simply multiply that volume by 3.375 to find the volume of the new, scaled shape. For instance, if the original shape has a volume of 10 cubic units, the volume of the similar shape with a scale factor of 1.5 would be 10 × 3.375 = 33.75 cubic units.

This principle applies to any similar shapes, whether they are cubes, spheres, or more complex geometrical figures. Always remember that the scale factor for volume is the cube of the linear scale factor.