Lesson 37: Areas of Combined Polygons

New Concept

This course introduces foundational math skills, showing how to break down complex problems into simpler, manageable steps. Mastering basics prepares you for advanced challenges.

What’s next

Next, you’ll apply this strategy in worked examples, learning to calculate the area of combined polygons by dividing or subtracting simpler shapes.

Property

The area of some polygons can be found by dividing the polygon into smaller parts and finding the area of each part.

Examples

A trapezoid is split into a rectangle and a triangle. Area = $(7 \cdot 5) + \frac{1}{2}(2 \cdot 5) = 35 + 5 = 40 \text{ cm}^2$.
An L-shaped figure is split into two rectangles. Area = $(8 \text{ ft} \cdot 6 \text{ ft}) + (4 \text{ ft} \cdot 4 \text{ ft}) = 48 + 16 = 64 \text{ ft}^2$.

Explanation

Think of a weirdly shaped polygon as a puzzle! You can't solve it all at once. So, you break it into easy shapes you already know, like rectangles and triangles. Find the area of each piece, add them all up, and voilà! You have solved the big puzzle and found the total area.

Property

When a polygon contains a right triangle with an unknown side, use the Pythagorean Theorem ($a^2 + b^2 = c^2$) to find the missing length before calculating the area.

Examples

A shape has a right triangle piece with a hypotenuse of $13$ ft and one leg of $5$ ft. $x^2 + 5^2 = 13^2$, so the missing leg $x$ is $12$ ft.
To find the area of a right triangle with leg $a$, leg $8$, and hypotenuse $10$: first find $a^2 + 8^2 = 10^2 \implies a = 6$. Area = $\frac{1}{2}(6)(8) = 24 \text{ cm}^2$.

Explanation

Sometimes, a side length is playing hide-and-seek! If your shape has a right triangle, you can use the magic of the Pythagorean Theorem to find that missing length. Once you have found it, you can go back to calculating the area of the pieces like a true math detective. No mystery can stop you!

Property

Sometimes, it is easier to find the area of a composite figure by imagining it as a larger shape with a piece missing.
Total Area = Area of the Large Shape - Area of the Missing Piece (Hole)

Examples

• The "C" Shape: A C-shaped figure can be imagined as a large 12 by 10 rectangle (Area = 120) that has a 6 by 4 rectangular hole cut out of it (Area = 24). The total area of the "C" shape is 120 - 24 = 96 square units.
• Square with a Hole: A large 12 by 12 square has a smaller 5 by 5 square removed from its center. The remaining area is 144 - 25 = 119 square units.

Explanation

Imagine baking a giant, perfect rectangular cookie, and then using a smaller cookie cutter to take a bite out of it! To find the area of your leftover cookie, you find the area of the whole cookie first, and then subtract the area of the piece you removed. This trick is extremely helpful for shapes that have holes or empty spaces in the middle.