Area and Perimeter of Any Shape: Complete Formula Guide

Area and perimeter come up everywhere, from 3rd-grade worksheets to the SAT to laying flooring in your living room. The math is not complicated, but the formulas are easy to mix up, and the units trip students up constantly. This guide gives you every formula, when to use it, and how to handle the tricky shapes.

Free tool: try the problems from this lesson in our Area & Perimeter Calculator.

The Two Big Ideas

Perimeter is the distance around a shape. It is measured in linear units: inches, cm, feet, meters.

Area is the space inside a shape. It is measured in square units: in², cm², ft², m².

If you ever confuse them, picture a fence (perimeter) vs. The grass inside (area).

Squares and Rectangles

Square (all sides equal, length \(s\))

• Perimeter: \(P = 4s\)
• Area: \(A = s^2\)

Rectangle (length \(l\), width \(w\))

• Perimeter: \(P = 2l + 2w\)
• Area: \(A = l \cdot w\)

Example: A rectangle 8 cm by 5 cm. \(P = 2(8) + 2(5) = 26\) cm. \(A = 8 \times 5 = 40\) cm².

Triangles

General triangle (base \(b\), height \(h\))

• Area: \(A = \dfrac{1}{2} b h\)

The height must be perpendicular to the base, not just any side.

Perimeter of any triangle

Add the three sides.

Special triangles

Right triangle: Use the legs \(a\) and \(b\) as base and height. \(A = \dfrac{1}{2} ab\). The hypotenuse \(c\) satisfies \(a^2 + b^2 = c^2\).

Equilateral triangle (all sides equal length \(s\)): – \(P = 3s\) – \(A = \dfrac{\sqrt{3}}{4} s^2\)

Heron’s formula (when you know all three sides \(a, b, c\) but not the height):

\(s = \dfrac{a + b + c}{2}\)

\(A = \sqrt{s(s-a)(s-b)(s-c)}\)

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Circles

For a circle with radius \(r\) (or diameter \(d = 2r\)):

• Circumference (perimeter): \(C = 2\pi r = \pi d\)
• Area: \(A = \pi r^2\)

Use \(\pi \approx 3.14\) or \(\pi \approx \dfrac{22}{7}\), or leave the answer in terms of \(\pi\) when the problem allows.

Example: A circle with radius 7 cm. \(C = 2\pi(7) = 14\pi \approx 43.98\) cm. \(A = \pi(7)^2 = 49\pi \approx 153.94\) cm².

Parallelograms and Trapezoids

Parallelogram (base \(b\), perpendicular height \(h\))

• Area: \(A = b \cdot h\)
• Perimeter: \(P = 2(\text{base} + \text{slant side})\)

Trapezoid (parallel sides \(a\) and \(b\), height \(h\))

• Area: \(A = \dfrac{1}{2}(a + b) h\)
• Perimeter: add all four sides.

Rhombus (all sides equal length \(s\), diagonals \(d_1, d_2\))

• Perimeter: \(P = 4s\)
• Area: \(A = \dfrac{1}{2} d_1 d_2\)

Regular Polygons (5+ Sides)

For a regular polygon with \(n\) sides of length \(s\) and apothem (perpendicular distance from center to a side) \(a\):

• Perimeter: \(P = ns\)
• Area: \(A = \dfrac{1}{2} \cdot P \cdot a = \dfrac{1}{2} \cdot ns \cdot a\)

The apothem can be found using trigonometry: \(a = \dfrac{s}{2 \tan(180° / n)}\).

Composite Shapes (the SAT/ACT Favorite)

Many test questions give you an L-shape, a shape with a circle cut out, or a triangle plus a rectangle.

Strategy: Break the shape into pieces you already know, find the area of each piece, then add or subtract.

Example: A rectangle 10 cm by 8 cm with a circle of radius 3 cm cut out of it.

Rectangle area = $80$ cm². Circle area = \(\pi(3)^2 = 9\pi \approx 28.27\) cm². Remaining area = \(80 – 28.27 \approx 51.73\) cm².

3D Versions (Surface Area)

If a test asks for surface area, it is the sum of the areas of all the faces. You still use the 2D formulas above; you just add them up.

For example, a rectangular prism with length \(l\), width \(w\), height \(h\):

\[\text{Surface Area} = 2(lw + lh + wh)\]

Common Mistakes

Forgetting to square the units

Area is in square units. Writing “40 cm” for area is wrong. Write “40 cm².”

Using the wrong height

Triangles and parallelograms need the perpendicular height, not the slant side.

Confusing radius and diameter

\(d = 2r\). If a problem gives you a diameter of 14 cm, the radius is 7 cm.

Forgetting \(\pi\)

Some students compute area as \(r^2\) without the \(\pi\).

Mixing units

A rectangle with sides “12 inches and 1 foot” needs to be converted: both in inches gives \(12 \times 12 = 144\) in², or both in feet gives \(1 \times 1 = 1\) ft².

Mis-reading composite figures

Always re-draw composite shapes and label every measurement before computing.

Practical Examples

Painting a wall

A wall is 14 ft long and 9 ft tall. There is a window 4 ft by 3 ft. How much wall needs painting?

Wall area: \(14 \times 9 = 126\) ft². Window area: \(4 \times 3 = 12\) ft². Paintable area: \(126 – 12 = 114\) ft².

If a gallon of paint covers 350 ft², you need less than a gallon. Buy one.

Building a fence

You want to fence in a yard that is 60 ft by 40 ft. How much fencing?

Perimeter: \(2(60) + 2(40) = 120 + 80 = 200\) ft.

Pizza math

A 16-inch pizza vs. Two 10-inch pizzas, which gives more pizza?

16-inch: radius 8, area \(= 64\pi \approx 201\) in². Two 10-inch: each radius 5, area \(= 25\pi \approx 78.5\) in² each → 157 in² total.

The 16-inch pizza is the better deal.

Free Resources

Effortless Math has a full free geometry library:

• Geometry Worksheets (resource currently unavailable), area, perimeter, and more.
• Math Topics Library, every geometry topic explained.
• Geometry eBooks, full geometry workbooks.

Frequently Asked Questions

How do you find the area and perimeter of a shape? Perimeter is the distance around a shape, so add the lengths of all its sides. Area is the space inside, found with the formula for that shape. For a rectangle 8 cm by 5 cm, the perimeter is 2(8) + 2(5) = 26 cm and the area is 8 × 5 = 40 cm². For a triangle, area is ½ × base × height, and for a circle it is πr².

What is the area and perimeter of a square? For a square with side length s, the perimeter is P = 4s and the area is A = s². A square with 6-inch sides has a perimeter of 4 × 6 = 24 inches and an area of 6² = 36 square inches. Keep the units straight: perimeter uses linear units like inches, while area uses square units like in².

How do you find the area and perimeter of an irregular shape? Split the irregular shape into rectangles, triangles or other simple shapes, find the area of each piece, and add them together, subtracting any cut-out parts. For perimeter, add only the outside edges, not the lines you drew inside. For example, an L-shape made from a 6 × 4 rectangle and a 3 × 2 rectangle has an area of 24 + 6 = 30 square units.

Is area and perimeter the same thing? No. Perimeter measures the distance around a shape in linear units, such as feet, while area measures the space inside in square units, such as ft². Picture a yard: the fence is the perimeter and the grass is the area. Two rectangles can share a perimeter but differ in area: 2 × 8 and 5 × 5 both have a perimeter of 20, but their areas are 16 and 25.

Can a shape have the same area and perimeter? Yes, the numbers can match, even though the units differ because area is in square units. A 4 × 4 square has a perimeter of 16 and an area of 16. A 3 × 6 rectangle works too: its perimeter is 2(3) + 2(6) = 18 and its area is 3 × 6 = 18. If you switch units, say from feet to inches, the numbers will no longer match.

What is the perimeter of any shape? For any polygon, the perimeter is the sum of all its side lengths, so a triangle with sides 5, 7 and 9 has a perimeter of 21. Some shapes have shortcuts: a square is 4s, a rectangle is 2l + 2w, and an equilateral triangle is 3s. A circle’s perimeter is called its circumference, C = 2πr, so a circle with radius 5 has C = 10π, about 31.4.

Master the Formulas. Master the Test.

Area and perimeter problems are point-for-point gifts on any standardized test, if you know your formulas cold. Print a formula sheet. Drill 10 problems a day for a week. By next Sunday, you will solve any area or perimeter problem in 60 seconds or less.

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