Area and Perimeter — Formulas, Examples, and Practice

Grade: 6-8 | Topic: Geometry

What You Will Learn

By the end of this guide you will be able to calculate the perimeter and area of the most common 2D shapes — rectangles, squares, triangles, parallelograms, trapezoids, and circles. You will understand when to use each formula, how the units differ between perimeter and area, and how to apply these skills to real-world measurement problems.

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:

$P = s_1 + s_2 + s_3 + \cdots + s_n$

where $s_1, s_2, \ldots, s_n$ are the lengths of the $n$ sides.

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

$P_{\text{rectangle}} = 2l + 2w$

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 $s$):

$P_{\text{square}} = 4s$

Area: measuring the space inside

The area of a shape is the amount of flat surface it covers. Area is measured in square units (cm$^2$, m$^2$, ft$^2$, etc.).

Rectangle:

$A_{\text{rectangle}} = l \times w$

A rectangle that is 8 cm by 5 cm covers $8 \times 5 = 40$ cm$^2$.

Square:

$A_{\text{square}} = s^2$

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

$A_{\text{triangle}} = \frac{1}{2} \times b \times h$

where $b$ is the length of the base and $h$ is the perpendicular height (the vertical distance from the base to the opposite vertex). The height must be measured at a right angle to the base — not along one of the sides.

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

$A_{\text{parallelogram}} = b \times h$

Here $h$ is the perpendicular height, not the length of the slanted side.

Trapezoid: A trapezoid (trapezium in British English) has one pair of parallel sides called bases ($b_1$ and $b_2$). Its area is:

$A_{\text{trapezoid}} = \frac{1}{2}(b_1 + b_2) \times h$

This formula works because a trapezoid is essentially the average of its two bases multiplied by the height.

Circles: circumference and area

Circles do not have straight sides, so instead of "perimeter" we use the term circumference. Both formulas involve the constant $\pi \approx 3.14159$:

$C = 2\pi r = \pi d$

$A_{\text{circle}} = \pi r^2$

where $r$ is the radius (distance from the center to the edge) and $d = 2r$ is the diameter. For example, a circle with radius 7 cm has circumference $2\pi(7) = 14\pi \approx 43.98$ cm and area $\pi(7)^2 = 49\pi \approx 153.94$ cm$^2$.

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. $P = 2l + 2w = 2(12) + 2(7) = 24 + 14 = 38 \text{ m}$

Step 3: Calculate the area. $A = l \times w = 12 \times 7 = 84 \text{ m}^2$

Answer: Perimeter = 38 m, Area = 84 m$^2$

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. $A = \frac{1}{2} \times b \times h$

Step 2: Substitute the values. $A = \frac{1}{2} \times 10 \times 6 = \frac{60}{2} = 30 \text{ cm}^2$

Answer: 30 cm$^2$

Example 3: Circumference and area of a circle (medium)

Problem: A circular pond has a diameter of 10 m. Find its circumference and area. Use $\pi \approx 3.14$.

Step 1: Find the radius. $r = \frac{d}{2} = \frac{10}{2} = 5 \text{ m}$

Step 2: Calculate the circumference. $C = 2\pi r = 2 \times 3.14 \times 5 = 31.4 \text{ m}$

Step 3: Calculate the area. $A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ m}^2$

Answer: Circumference = 31.4 m, Area = 78.5 m$^2$

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. $b_1 = 8$ cm, $b_2 = 14$ cm, $h = 5$ cm.

Step 2: Apply the trapezoid area formula. $A = \frac{1}{2}(b_1 + b_2) \times h = \frac{1}{2}(8 + 14) \times 5$

Step 3: Simplify. $A = \frac{1}{2}(22) \times 5 = 11 \times 5 = 55 \text{ cm}^2$

Answer: 55 cm$^2$

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. $A_{\text{rect}} = l \times w = 4 \times 3 = 12 \text{ m}^2$

Step 2: Calculate the area of the triangular section. $A_{\text{tri}} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 4 \times 1.5 = 3 \text{ m}^2$

Step 3: Add both areas together. $A_{\text{total}} = 12 + 3 = 15 \text{ m}^2$

Answer: 15 m$^2$

Common Mistakes

Mistake 1: Confusing area and perimeter formulas

❌ Area of a rectangle = $2l + 2w$ (this is the perimeter formula)

✅ Area of a rectangle = $l \times w$

Why this matters: Perimeter adds lengths and gives a result in linear units (cm, m). Area multiplies dimensions and gives a result in square units (cm$^2$, m$^2$). Always check whether your answer should be in units or square units — that tells you which formula to use.

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

❌ A triangle with sides 10, 8, and 6 — using 8 as the height with base 10: $A = \frac{1}{2} \times 10 \times 8 = 40$

✅ The height must be measured perpendicular to the base. If the base is 10 and the perpendicular height is 6: $A = \frac{1}{2} \times 10 \times 6 = 30$

Why this matters: The height in area formulas must always form a 90-degree angle with the base. Using a slant side overestimates the area because the slant side is longer than the vertical distance.

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

❌ $A = \pi r = 3.14 \times 5 = 15.7$

✅ $A = \pi r^2 = 3.14 \times 25 = 78.5$

Why this matters: The formula $\pi r$ gives only half the circumference, not the area. Area requires $r^2$ because you are measuring two-dimensional space. Missing the exponent dramatically underestimates the result.

Practice Problems

Try these on your own before checking the answers:

1. Find the perimeter and area of a rectangle that is 15 cm long and 9 cm wide.
2. A triangle has a base of 12 m and a perpendicular height of 8 m. What is its area?
3. A circle has a radius of 4 cm. Find its circumference and area (use $\pi \approx 3.14$).
4. A trapezoid has parallel sides of 6 cm and 10 cm, and a height of 4 cm. Find its area.
5. A room is shaped like a rectangle (5 m by 4 m) with a semicircular alcove on one short side (diameter 4 m). Find the total floor area (use $\pi \approx 3.14$).

Click to see answers

1. $P = 2(15) + 2(9) = 48$ cm. $A = 15 \times 9 = 135$ cm$^2$.
2. $A = \frac{1}{2} \times 12 \times 8 = 48$ m$^2$.
3. $C = 2 \times 3.14 \times 4 = 25.12$ cm. $A = 3.14 \times 16 = 50.24$ cm$^2$.
4. $A = \frac{1}{2}(6 + 10) \times 4 = \frac{1}{2}(16) \times 4 = 32$ cm$^2$.
5. Rectangle: $5 \times 4 = 20$ m$^2$. Semicircle: $\frac{1}{2} \times 3.14 \times 2^2 = \frac{1}{2} \times 3.14 \times 4 = 6.28$ m$^2$. Total: $20 + 6.28 = 26.28$ m$^2$.

Summary

• Perimeter is the distance around a shape, measured in linear units. Add all side lengths, or use $P = 2l + 2w$ for rectangles.
• Area is the space inside a shape, measured in square units. Multiply dimensions (length times width, base times height, etc.).
• For triangles, the area formula uses half the base times the perpendicular height: $A = \frac{1}{2}bh$.
• For circles, circumference is $C = 2\pi r$ and area is $A = \pi r^2$. Always square the radius for area.
• Composite shapes can be broken into familiar parts — calculate each area separately, then add them together.

Related Topics

• Area of a Triangle — Formula and Examples
• Area and Circumference of a Circle — Formulas and Examples
• How to Find the Area of a Trapezoid
• How to Find the Perimeter of Any Polygon
• Surface Area of 3D Shapes — Prisms, Cylinders, and More

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