## What You Will Learn
**Perimeter** is the total distance around a 2D shape's boundary; **area** is the amount of surface it covers. Perimeter uses linear units (cm, m); area uses square units (cm², m²). This guide covers formulas for rectangles, squares, triangles, parallelograms, trapezoids, and circles, plus worked examples and real-world applications for grades 6-8.

## Theory
### Perimeter: measuring the boundary
The **perimeter** of a shape is the total length of its boundary — the distance you would walk if you traced all the way around the outside. Perimeter is measured in **linear units** (cm, m, ft, etc.).

For any polygon (a shape with straight sides), you find the perimeter by adding up all the side lengths:

P=s1+s2+s3+⋯+sn

where s1,s2,…,sn are the lengths of the n sides.

**Rectangle:** A rectangle has two pairs of equal sides (length l and width w), so:

Prectangle=2l+2w

For example, a rectangle that is 8 cm long and 5 cm wide has a perimeter of 2(8)+2(5)=16+10=26 cm.

**Square:** All four sides are equal (side length s):

Psquare=4s

### Area: measuring the space inside
The **area** of a shape is the amount of flat surface it covers. Area is measured in **square units** (cm², m², ft², etc.).

**Rectangle:**

Arectangle=l×w

A rectangle that is 8 cm by 5 cm covers 8×5=40 cm².

**Square:**

Asquare=s²

**Triangle:** A triangle's area is half the area of the rectangle that encloses it:

Atriangle=12×b×h

where b is the length of the base and h is the perpendicular height.

**Parallelogram:** A parallelogram can be rearranged into a rectangle with the same base and height:

Aparallelogram=b×h

**Trapezoid:** A trapezoid has one pair of parallel sides called bases (b1 and b2). Its area is:

Atrapezoid=12(b1+b2)×h

### Circles: circumference and area
Circles do not have straight sides, so instead of "perimeter" we use the term **circumference**.

C=2πr=πd

Acircle=πr²

where r is the radius and d=2r is the diameter. For example, a circle with radius 7 cm has circumference 2π(7)≈43.98 cm and area π(7)²≈153.94 cm².

## Worked Examples
### Example 1: Perimeter and area of a rectangle (easy)
**Problem:** A garden is 12 m long and 7 m wide. Find its perimeter and area.

**Step 1:** Identify the dimensions.
Length l=12 m, width w=7 m.

**Step 2:** Calculate the perimeter.
P=2l+2w=2(12)+2(7)=38 m

**Step 3:** Calculate the area.
A=l×w=12×7=84 m²

**Answer:** Perimeter = **38 m**, Area = **84 m²**

### Example 2: Area of a triangle (medium)
**Problem:** A triangle has a base of 10 cm and a perpendicular height of 6 cm. Find its area.

**Step 1:** Write down the formula for the area of a triangle.
A=12×b×h

**Step 2:** Substitute the values.
A=12×10×6=30 cm²

**Answer:** **30 cm²**

### Example 3: Circumference and area of a circle (medium)
**Problem:** A circular pond has a diameter of 10 m. Find its circumference and area.

**Step 1:** Find the radius.
r=d/2=5 m

**Step 2:** Calculate the circumference.
C=2πr=31.4 m

**Step 3:** Calculate the area.
A=πr²=78.5 m²

**Answer:** Circumference = **31.4 m**, Area = **78.5 m²**

### Example 4: Area of a trapezoid (medium)
**Problem:** A trapezoid has parallel sides of 8 cm and 14 cm, and a height of 5 cm. Find its area.

**Step 1:** Identify the bases and height.
b1=8 cm, b2=14 cm, h=5 cm.

**Step 2:** Apply the trapezoid area formula.
A=12(b1+b2)×h=55 cm²

**Answer:** **55 cm²**

### Example 5: Composite shape — combining rectangle and triangle (challenging)
**Problem:** A wall has the shape of a rectangle (4 m wide, 3 m tall) with a triangular roof on top (base 4 m, height 1.5 m). Find the total area of the wall.

**Step 1:** Calculate the area of the rectangular section.
Arect=l×w=12 m²

**Step 2:** Calculate the area of the triangular section.
Atri=12×b×h=3 m²

**Step 3:** Add both areas together.
Atotal=15 m²

**Answer:** **15 m²**

## Common Mistakes
**Mistake 1: Confusing area and perimeter formulas**

Area of a rectangle = l×w. Always check whether your answer should be in units or square units.

**Mistake 2: Using the slant side instead of the perpendicular height.**

Height must always form a 90-degree angle with the base.

**Mistake 3: Forgetting to square the radius when finding the area of a circle.**

Area requires r² because you are measuring two-dimensional space.