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.