An **inscribed rectangle** is a rectangle drawn within a shape. In calculus, we’re mostly concerned with the *largest* inscribed rectangle; The largest that doesn’t break through the edges of the shape. A **circumscribed rectangle** is drawn outside of a shape; Usually we want to know the smallest rectangle that can completely enclose a shape.

## Use of Inscribed and Circumscribed Rectangles in Calculus

Inscribed rectangles and circumscribed rectangles are used in a few different areas in calculus, most notably in Riemann Sums and Optimization.

## 1. Riemann Sums: Use of Circumscribed and Inscribed Rectangle

## Inscribed rectangles and circumscribed rectangles in Riemann Sums

The area under a curve can be approximated with a series of rectangles:

**Inscribed rectangles**are an under-approximation of the area (called a lower sum).**Circumscribed rectangles**are an over-approximation of the area (called an upper sum).

For step by step examples and how-to video, see: Riemann sums.

## 2. Optimization: Largest Inscribed Rectangle Within a Circle

Many optimization problems (which deal with the “biggest” or “smallest”) use inscribed and circumscribed rectangles.

**Example**: What is the largest inscribed rectangle that can fit into a circle with a radius of 1?

Step 1: **Formulate a function to maximize**.

We know that the diagonal of any inscribed rectangle (blue line) has a length of 2 (because 2 * radius = diameter). We can use this information and the Pythagorean theorem (a^{2} + b^{2} = c^{2}) to get:

**w ^{2} + h^{2} = 4.**

We need to get this formula in terms of one variable to create our function:

- Solve for w: w
^{2}+ h^{2}= 4 → w = √(4 – h)^{2}. - Substitute (1) into A = w * h → A = h * √(4 – h
^{2}).

The formula we need to maximize is A = h * √(4 – h^{2}).

Step 2: **Graph the function** (I used Desmos.com). We want to find the function’s maximum height.

The function has a maximum x-value (actually, our “h”) at x = √2 ≈ 1.414. Therefore, the largest rectangle has a height of √2.

If h = √2, then, putting that into the Pythagorean theorem:

1.414^{2} + b^{2} = 4

2 + b^{2} = 4

2 + √2,^{2} = 4

2 + 2 = 4

The inscribed rectangle is a 2 x 2 square with an area of 4.