When studying the topic of "area", a geometry question that comes up frequently is how to find the area of a non-standard shape. In this post, we'll solve a common example - finding the area of a heart shape.

A "heart" is not a well defined description. We have wide hearts like this -

And narrow hearts like this:

So when solving this problem, unless we have some specific measurements or instructions that say otherwise, we will need to make some simplifying assumptions.

## Problem

In the shape below, D is the length between its bottom tip and the bottom of the indent between the rounded areas. Find the area of the "heart shape."

## Strategy

In this case, we will assume that the area of a "heart" shape is made up of a square with two semicircles attached to its sides. then, we will use the strategy of breaking down a complex shape into the simple shapes that make up its parts.

With that assumption, we have a square, whose area is given by the formula A_{square}=a^{2}, and two semicircles. The distance D is simply the square's diagonal.

The area of each semicircle is given by the formula A_{semicircle}=π*r^{2}/2. And in our case, the diameter is simply the side of the square, a, so r=a/2

To find the length of the side, a, from the diagonal, D, we divide by √2, as we have shown here - How to find the side of a square from its diagonal.

So, the area of the square is a^{2}= (D/√2)^{2}=D^{2}/2. The area of each semicircle is π*r^{2}/2= π*(D/2√2)^{2}/2, and we have two of them, so together, the area of the two semicircles is π*(D/2√2)^{2}=π*D^{2}/8

And the area of the heart shape is A_{square}+2A_{semicircle} = D^{2}/2+π*D^{2}/8=(1+π/4)D^{2}/2