Why Do We Measure Shapes?
People have been measuring shapes for thousands of years. Ancient farmers needed to know how much land they owned. Builders needed to figure out how much material to use for walls and floors. These everyday problems led to the math we now call geometry (the study of shapes, sizes, and spaces).
Two of the most useful measurements are perimeter (the total distance around a shape) and area (the amount of flat space a shape covers). Let's look at how these ideas developed over time.
The big question that geometry answers is: How do we describe the size of a flat shape using numbers? Perimeter and area give us two different ways to do exactly that.
Core Principles & Definitions
Before we jump into formulas, you need to understand a few key ideas. These principles apply to every shape you will work with.
Perimeter = Distance Around
Area = Space Inside
Units Matter
Know Your Dimensions
Seeing Perimeter and Area
The diagram below shows a rectangle, a triangle, and a circle side by side. The colored border represents the perimeter (or circumference) of each shape, and the shaded interior represents the area.
Notice how each shape uses a different formula. The rectangle multiplies length times width. The triangle uses half the base times the height. The circle uses π (pi), which is about 3.14. In the next section, we will list every formula you need.
The Formulas You Need
Here are the key formulas for the shapes you will see on the TACHS. For each one, we show both the perimeter (or circumference) and the area formula.
Rectangle
Square
Triangle
Circle
Parallelogram & Trapezoid
Shape-by-Shape Guide
The diagram below puts all six shapes on one page with their formulas. Use it as a quick-reference sheet when you practice.
| Shape | Perimeter Formula | Area Formula |
|---|---|---|
| Square | P = 4s | A = s² |
| Rectangle | P = 2l + 2w | A = l × w |
| Triangle | P = a + b + c | A = ½ × b × h |
| Circle | C = 2πr or πd | A = πr² |
| Parallelogram | P = 2a + 2b | A = b × h |
| Trapezoid | P = a + b₁ + c + b₂ | A = ½(b₁ + b₂) × h |
Worked Example: A Backyard Garden
Maria wants to put a fence around her rectangular garden and then fill it with soil. The garden is 12 feet long and 8 feet wide. How much fencing does she need (perimeter)? How much soil will cover the garden (area)?
Common Mistakes & How to Avoid Them
Many students mix up formulas or forget steps. The table below shows the most common errors and how to fix them.
| Mistake | Why It's Wrong | Fix |
|---|---|---|
| Using the slanted side as the height of a triangle or parallelogram | The height must be perpendicular (at a 90° angle) to the base | Look for a dashed line with a small square at its base — that marks the true height |
| Forgetting to halve the triangle area | A = b × h gives you a rectangle, not a triangle | Always multiply by ½ (or divide your answer by 2) |
| Using diameter instead of radius in A = πr² | If d = 10, then r = 5, not 10. Using 10 gives you 4× the correct area | If given the diameter, divide by 2 first to get the radius |
| Writing area in regular units (cm instead of cm²) | Area measures square units. The answer needs a ² symbol | Double-check: perimeter → regular units; area → squared units |
| Confusing perimeter and area formulas | Adding when you should multiply (or vice versa) | Remember: Perimeter = add sides; Area = multiply dimensions |
Connecting to Advanced Topics
Once you master perimeter and area of flat shapes, you are ready for bigger ideas. Here is a preview of what comes next.
| What You Know Now | What Comes Next |
|---|---|
| Perimeter of flat shapes | Surface area of 3D shapes (boxes, cylinders, spheres) |
| Area of flat shapes | Volume of 3D shapes (how much space is inside) |
| Using π for circles | Volume of cylinders, cones, and spheres |
| Simple shapes | Composite figures (shapes made of two or more simple shapes combined) |
On the TACHS, you might see a problem that combines shapes. For example, you could be asked to find the area of an L-shaped room. The trick is to break it into two rectangles, find each area, and add them together. The formulas you learned today are the building blocks for those problems.
Practice Problems
Try these five problems. They get harder as you go. Check your work against the answers below each problem.
Lesson Summary
Perimeter is the distance around a shape, found by adding all the side lengths. For a rectangle, use P = 2l + 2w. For a square, use P = 4s. For a circle, the perimeter is called the circumference: C = 2πr. Perimeter is measured in regular units like cm, ft, or m.
Area is the space inside a shape, found by multiplying dimensions. A rectangle's area is A = l × w. A triangle's area is A = ½ × b × h. A circle's area is A = πr². For a trapezoid, use A = ½(b₁ + b₂) × h. Area is always in square units like cm², ft², or m². Always identify your shape, pick the right formula, plug in the numbers, and label your answer with the correct unit.