# Length and Area

Geometry is the study of shapes and sizes. Having learnt about angles and shapes, we now move on to length and area. By measuring the physical dimensions of an object, we can understand how length, area and volume relate to each other. For example, the size of the sun, the size of a room, and the space taken up by an aircraft can be expressed in terms of a measurement.

Here are some tips for you to get started:

- When in doubt, draw the diagram for yourself.
- Label various lengths and areas that you know.
- Keep an eye out for similar and congruent figures.
- Try viewing the problem from a different perspective.
- Make an assumption, which would help you form an educated guess.

Within a large square, we inscribe a circle.

This circle circumscribes another square.What is the ratio of the area of the smaller square to the larger square?

\[ A)\, 1:2 \quad B) \, 1:3 \quad C) \, 1:4 \quad D) \, 1:6 \]

There doesn't seem to be an easy way to start. As it turns out, if we view the problem from a different perspective, we have a short solution:

If we rotate the smaller square by \( 45 ^ \circ \) we get the following diagram:

It is now obvious that the smaller square is half the area of the larger square. Hence, the ratio of their areas is \( 1 : 2 \).

For a more standard solution, let's draw the diagram and label various lengths:

Suppose that the large square has side length 2. Then, the circle has a diameter of 2, which is equal to the diagonal of the smaller square. Hence, the smaller square has a side length of \( \frac{2}{ \sqrt{2} } = \sqrt{2} \). As such, the ratio of their areas is \( 2^2 : \sqrt{2} ^2 = 4 : 2 = 2 : 1 \).

Geometry problems involving length and area can be classified as:

- Perimeter: Simply find the sum of all of the side lengths! Sometimes, you have to be crafty to determine the individual sides.
- Area of common shapes: triangle, rectangle and circle.
- Composite figures: These comprise of several basic shapes put together, and we have to use our creativity to determine the shaded area / perimeter.
- Similar Polygons: Knowing the ratio of similarity, we can deduce the ratio of their perimeters and areas.

**Cite as:**Length and Area.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/lively-length-and-area/