What is the difference between area and perimeter?

Perimeter is the total distance around the outside of a shape (measured in units), while area is the amount of surface the shape covers (measured in square units).

What is the area formula for a triangle?

Area = (base × height) ÷ 2. The height must be perpendicular (at a right angle) to the base.

How do you find the area of a circle?

Area = π × r², where r is the radius (half the diameter). Use π ≈ 3.14159 or 22/7 as an approximation.

Does a square have the same area and perimeter formula as a rectangle?

A square uses the same formulas but with all four sides equal: Area = s² and Perimeter = 4s, where s is the side length.

What units are used for area vs perimeter?

Perimeter uses linear units (cm, m, ft, in). Area uses square units (cm², m², ft², in²). Always include the correct unit in your answer.

Area and Perimeter: Formulas, Examples & Practice

The Big Picture: Area vs. Perimeter

Imagine you want to put a fence around your backyard and then lay sod inside it. The length of fence you need is the perimeter. The amount of sod you need is the area. Both measurements describe the same shape, but they answer completely different questions.

Perimeter is a one-dimensional measurement — a total length, expressed in cm, m, ft, etc. Area is two-dimensional — it counts square units (cm², m², ft²). Confusing the two is the most common error in geometry, so keeping this distinction sharp is worth the effort.

Step 1: Rectangles and Squares

STEP 1

The rectangle is the foundation. Every other shape's formula connects back to it in some way.

Rectangle

Perimeter = 2(l + w)
Area = l × w

Square (l = w = s)

Perimeter = 4s
Area = s²

Example: A rectangle 8 cm long and 5 cm wide.
Perimeter = 2(8 + 5) = 2 × 13 = 26 cm
Area = 8 × 5 = 40 cm²

Remember: length and width must be in the same unit before you calculate. If l = 2 m and w = 50 cm, convert 50 cm = 0.5 m first.

Step 2: Triangles

STEP 2

A triangle's perimeter is simply the sum of its three sides. Its area uses a base and a perpendicular height.

Area = (base × height) ÷ 2

The height must form a right angle (90°) with the base — not the slanted side of the triangle. For right triangles, the two legs serve as base and height directly. For acute or obtuse triangles, you may need to draw the height outside the triangle.

Example: Triangle with base 10 m, perpendicular height 6 m, and sides 8 m, 8 m, 10 m.
Perimeter = 8 + 8 + 10 = 26 m
Area = (10 × 6) ÷ 2 = 60 ÷ 2 = 30 m²

Why ÷ 2? A triangle is exactly half a rectangle with the same base and height. Picture cutting a rectangle diagonally — each half is a triangle.

Step 3: Parallelograms and Trapezoids

STEP 3

Parallelogram

Area = base × perpendicular height
(same logic as rectangle, slant doesn't add area)

Trapezoid (Trapezium)

Area = ½ × (a + b) × h
where a, b are parallel sides and h is perpendicular height

Trapezoid example: Parallel sides 7 cm and 11 cm, perpendicular height 4 cm.
Area = ½ × (7 + 11) × 4 = ½ × 18 × 4 = ½ × 72 = 36 cm²

The trapezoid formula averages the two parallel sides to find the "effective width," then multiplies by the height — same idea as a rectangle with an averaged width.

Step 4: Circles

STEP 4

Circles use π (pi) ≈ 3.14159, the ratio of a circle's circumference to its diameter. Two key measurements: radius r (center to edge) and diameter d = 2r (edge to edge through center).

Circumference (perimeter) = 2πr = πd
Area = πr²

Example: Circle with radius 5 cm.
Circumference = 2 × π × 5 = 10π ≈ 31.4 cm
Area = π × 5² = π × 25 ≈ 78.5 cm²

A common mnemonic: "Pie are squared" (πr² = Area) and "Cherry pie's delicious" (C = πd). Leave your answer in terms of π when the problem doesn't specify a decimal approximation.

Step 5: Composite Figures

STEP 5

Real-world shapes are often combinations of simpler ones — an L-shaped room, a pool with semicircular ends, a garden with a triangular flowerbed. The strategy:

Decompose the composite shape into non-overlapping simple shapes.
Calculate the area of each piece separately.
Add or subtract to find the total (subtract when a piece is cut away).

Example — L-shape: An L-shaped floor. Treat it as a 10 × 8 rectangle minus a 4 × 3 rectangle cut from the top-right corner.
Large rectangle = 10 × 8 = 80 m²
Removed piece = 4 × 3 = 12 m²
Total area = 80 − 12 = 68 m²

For perimeter of composite figures, trace the outer boundary carefully — internal edges that were created by decomposing the shape are NOT part of the perimeter.

Step 6: Scaling — What Happens When You Resize a Shape?

STEP 6

Understanding how area and perimeter change when you scale a shape is essential for design and engineering:

If all sides are multiplied by a scale factor k, the perimeter multiplies by k.
The area multiplies by k².

Example: A 3 × 4 rectangle (Perimeter = 14, Area = 12) is scaled by k = 3.
New dimensions: 9 × 12
New Perimeter = 42 = 14 × 3 ✓
New Area = 108 = 12 × 9 = 12 × 3² ✓

This means doubling the size of a garden quadruples the amount of grass seed needed. This scaling relationship appears in architecture, mapmaking, and scientific modeling throughout professional life.

Practice Problems

Rectangle: A room is 6.5 m long and 4.2 m wide. Find the perimeter and area.
Solution: Perimeter = 2(6.5 + 4.2) = 2 × 10.7 = 21.4 m; Area = 6.5 × 4.2 = 27.3 m²
Triangle: Base = 14 cm, perpendicular height = 9 cm. Find the area.
Solution: Area = (14 × 9) ÷ 2 = 126 ÷ 2 = 63 cm²
Circle: A circular pond has a diameter of 10 m. Find its circumference and area. (Use π ≈ 3.14)
Solution: r = 5 m; Circumference = 2 × 3.14 × 5 = 31.4 m; Area = 3.14 × 25 = 78.5 m²
Composite: A shape is a rectangle 12 cm × 5 cm with a 3 cm × 3 cm square cut from one corner. Find the area.
Solution: Rectangle area = 60 cm²; Square removed = 9 cm²; Total = 51 cm²
Scaling: A square tile has side 4 cm. If the manufacturer creates a larger tile with side 12 cm, how many times greater is the new tile's area?
Solution: Scale factor k = 12/4 = 3; Area scales by k² = 9; New area = 16 × 9 = 144 cm² (vs. 16 cm²). The new tile is 9 times larger in area.

5 Common Mistakes to Avoid

Using the slant height instead of the perpendicular height for triangles. The area formula requires the height that is perpendicular to the base. Using a slant side gives a larger (wrong) answer.

Forgetting to square the radius in the circle area formula. Writing πr instead of πr² is extremely common. Always check: Area = π × r × r.

Mixing up diameter and radius. The diameter is the full width of a circle; the radius is half. Using d in the area formula (πd²) gives an answer 4 times too large.

Reporting area in linear units and perimeter in square units. Perimeter → cm, m, ft (linear). Area → cm², m², ft² (squared). This distinction signals whether you answered the right question.

Including internal edges when calculating composite-figure perimeter. When you split a shape to find area, the internal cuts are not part of the outer boundary. Trace the outside edge only.

Area and Perimeter