Why Do We Measure Area?
Thousands of years ago, people needed to measure land. Farmers had to know how much ground they owned. Tax collectors needed to figure out what people owed. The answer to all of these questions came down to one idea: area (the amount of flat space inside a shape).
Ancient civilizations discovered formulas for area step by step. They started with rectangles, then figured out triangles, parallelograms, and trapezoids. These same formulas still show up on tests like the SHSAT today!
The big question these formulas answer is simple: How much space does a flat shape cover? Let's learn the three formulas you need.
Core Principles of Area
Before jumping into formulas, you need to understand a few key ideas. These principles connect all three shapes we will study.
Area Is Measured in Square Units
Base and Height Must Be Perpendicular
Every Shape Connects to a Rectangle
Height Is Not Always a Side
Seeing the Three Shapes
The diagram below shows all three shapes side by side. Notice how each one labels the base (b) and the height (h). The dashed line is the height — it always makes a right angle with the base.
Look at the dashed lines carefully. In each shape, the height drops straight down at a 90° angle. The little square in the corner proves it. On the SHSAT, you might see shapes that are tilted or slanted. Don't let that trick you — always look for the perpendicular height.
The Three Area Formulas
Here are the three formulas you need to memorize. Each one is simple, but understanding why it works helps you remember it.
Why Each Formula Works
The best way to remember a formula is to understand the picture behind it. The diagram below shows how each shape connects to a simple rectangle.
In the top-left picture, the dashed rectangle has area b × h. The shaded triangle fills exactly half, so its area is ½ × b × h. In the top-right picture, the gold triangle on the left side of the parallelogram slides to the right. Now it is a rectangle with area b × h. The bottom picture shows that a trapezoid acts like a shape whose width is the average of its two bases. So the area is ½ × (b₁ + b₂) × h.
Worked Examples
Comparing the Three Formulas
All three formulas use a base and height, but each shape has its own twist. The table below helps you see the differences at a glance.
| Shape | Formula | Key Detail |
|---|---|---|
| Triangle | A = ½ × b × h | Don't forget the ½. Height is perpendicular to the base, not a slanted side. |
| Parallelogram | A = b × h | Same as a rectangle. Use the perpendicular height, NOT the slanted side. |
| Trapezoid | A = ½ × (b₁ + b₂) × h | Has TWO bases (the parallel sides). Add them first, then divide by 2, then multiply by the height. |
Connecting to Harder Topics
Once you master these three shapes, you will be ready for more advanced problems. On the SHSAT and in high school geometry, you will see composite figures (shapes made by combining simpler shapes). You might also work with shaded region problems where you subtract one area from another.
| What You Know Now | What Comes Next |
|---|---|
| Area of a single triangle | Break an irregular polygon into multiple triangles to find its total area |
| Area of a parallelogram | Find the area of rhombuses and other special parallelograms using diagonals |
| Area of a trapezoid | Use the trapezoid formula in coordinate geometry problems (with graphed vertices) |
| All three formulas | Surface area of 3-D solids (prisms, pyramids) that use triangle and trapezoid faces |
The key idea is that almost every area problem, no matter how complex, can be broken down into triangles, rectangles, parallelograms, and trapezoids. Mastering these building blocks makes harder problems much easier.
Practice Problems
Lesson Summary
You learned three essential area formulas. The area of a triangle is ½ × b × h because a triangle is half a rectangle. The area of a parallelogram is b × h because it rearranges into a rectangle. The area of a trapezoid is ½ × (b₁ + b₂) × h because you average its two parallel bases.
The most important rule across all three formulas: the height is always the perpendicular distance from the base to the opposite side — never a slanted edge. Watch for this on the SHSAT. With these three formulas, you can also tackle composite figure problems by breaking complex shapes into simpler pieces.