Basic Shapes: Perimeter & Area | Math Learning Guide

📋 Prerequisites

Geometry starts with the basics: perimeter and area of simple shapes. These formulas are the building blocks for everything else — composite figures, coordinate geometry, even 3D volume. Memorize them now and you'll save yourself countless headaches later.

📚 Key Concepts

What Are Perimeter and Area?

Perimeter: Total distance around a shape. Units: length (in, cm, etc.)
Area: Space inside a shape. Units: squared $(in^{2}, cm^{2},$ etc.)
Perimeter is like the fence around a yard; area is the grass inside it.

Basic Shape Formulas

Rectangle (length $l$, width $w$):

Perimeter: $ P = 2l + 2w = 2(l + w) $
Area: $ A = l \times w $

Square (side length $s$) — special case of rectangle:

Perimeter: $ P = 4s $
Area: $ A = s^2 $

Triangle (base $b$, height $h$):

Perimeter: sum of all three sides
Area: $ A = \frac{1}{2} \times b \times h $
The height must be perpendicular to the base (it's not just any side!)

Circle (radius $r$):

Circumference (perimeter of a circle): $ C = 2\pi r = \pi d $
Area: $ A = \pi r^2 $
Diameter: $ d = 2r $ (distance across the circle through center)
$ \pi \approx 3.14 $ or $ \frac{22}{7} $

Why the Formulas Work

Rectangle area: $If you have a 3 \times 5 grid, there are 3 rows of 5 squares = 15 total. That's why$ $ l \times w $.
Triangle area: A triangle is half of a rectangle with the same base and height. That's where the $ \frac{1}{2} $ comes from.
Circle area: Imagine cutting a circle into wedges and rearranging them into a "rectangle" with height $r$ and width $\pi r$. Area = $ \pi r \times r = \pi r^2 $.

Composite Figures (Cut & Paste)

Most competition problems aren't just one shape — they're combinations. Two strategies:

Addition (cut into pieces): Split the shape into simpler shapes, calculate each area, add them up.
Subtraction (negative space): Find the area of the big shape, subtract the missing parts.

Subtraction is often faster for shapes with holes or "cut-out" regions.

⚠️ Common Mistake: Using the wrong height for a triangle. The height must be perpendicular (at 90°) to the base. For an obtuse triangle, the height might fall outside the triangle itself.

💡 Key Insight: When you see a weird shape, don't panic. Ask yourself: can I cut this into rectangles and triangles? Or is it a big shape minus a smaller one? Almost every composite figure problem comes down to one of these two strategies.

✏️ Example Problems

📝 Example 1 (Triangle area)

A triangle has a base of 12 cm and a height of 5 cm. What is its area in square centimeters?

Formula: Area of triangle = $ \frac{1}{2} \times \text{base} \times \text{height} $

Plug in:

\[ A = \frac{1}{2} \times 12 \times 5 = \frac{1}{2} \times 60 = 30 \]

Answer: 30 $cm^{2}$

Tip: Multiply by $ \frac{1}{2} $ last — $it'\(s often easier to multiply base \times height first, then divide by 2.$\)

📝 Example 2 (Composite figure — subtraction)

$\(A 10 \times 10 square has a 3 \times 4 rectangle cut out from one corner. What is the area of the remaining shape$\)?

Strategy: Use subtraction — big area minus small area.

Step 1: Area of the big square:

\[ A_{\text{square}} = 10 \times 10 = 100 \]

Step 2: Area of the cut-out rectangle:

\[ A_{\text{rectangle}} = 3 \times 4 = 12 \]

Step 3: Subtract:

\[ A_{\text{remaining}} = 100 - 12 = 88 \]

Answer: 88