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Perimeter & Area
of Polygons

Master the formulas for perimeter and area of rectangles, triangles, parallelograms, trapezoids, and more.

SSAT Upper Level: Quantitative

What Is Perimeter?

The perimeter of a polygon is the total distance around its outside. You find it by adding up the lengths of every side.

Rectangle
\(8\)
\(5\)
 Pentagon
\(s_1\)
\(s_2\)
\(s_3\)
\(s_4\)
\(s_5\)

What Is Area?

Area measures the amount of two-dimensional space inside a polygon, expressed in square units (cm², ft², m²).

\(4 \text{ units}\)
\(3 \text{ units}\)
\(12 \text{ sq. units}\)

Why Perimeter & Area Matter

Perimeter in Life
Fencing a yard, framing a picture, or running around a track: all perimeter problems.
Area in Life
Painting a wall, laying carpet, or sizing a garden: you need area for every surface.
SSAT Relevance
Polygon questions appear on nearly every SSAT Quantitative section: know the formulas cold.

Formulas: Rectangle & Square

Rectangle
\(l\)
\(w\)
\( P = 2l + 2w \)
\( A = l \times w \)
 Square
\(s\)
\(s\)
\( P = 4s \)
\( A = s^2 \)

Formula: Triangle

\(b\)
\(h\)
\(a\)
\(c\)

Parallelogram & Trapezoid

Parallelogram
\(b\)
\(a\)
\(h\)
\( A = bh \)
 Trapezoid
\( b_1 \)
\( b_2 \)
\(h\)
\( A = \tfrac{1}{2}(b_1+b_2)h \)

Perimeter vs. Area: Key Distinction

Perimeter Area
Measures Distance around Space inside
Dimension 1-D (length) 2-D (surface)
Units \(\text{cm, ft, m}\) \(\text{cm}^2, \text{ft}^2, \text{m}^2\)
Common use Fencing, borders Flooring, painting
Operation Add side lengths Formula-based (e.g., \( \text{base} \times \text{height} \))

Step-by-Step Approach

1
Identify the shape. Is it a rectangle, triangle, parallelogram, trapezoid, or composite?
2
Decide: perimeter or area? Read the question for keywords like "fence" vs. "cover."
3
Write the formula. Pull the correct formula for that shape.
4
Plug in and compute. Substitute the given values, simplify, and include units.
5
Check your units. Perimeter → linear units. Area → square units.

Example 1: Rectangle

A garden is 12 ft long and 7 ft wide. Find its perimeter and area.

\(12 \text{ ft}\)
\(12 \text{ ft}\)
\(7 \text{ ft}\)
\(7 \text{ ft}\)

Example 2: Triangle

A triangle has sides 9 cm, 7 cm, and 10 cm. The height to the 10 cm base is 6 cm. Find the perimeter and area.

\(9\text{ cm}\)
\(7\text{ cm}\)
\(10\text{ cm}\)
\(6\text{ cm}\)

Example 3: Trapezoid

A trapezoid has parallel sides \( b_1 = \) 6 m and \( b_2 = \) 10 m, and height \( h = \) 4 m. Find its area.

\( b_1 = 6 \) m
\( b_2 = 10 \) m
\( h = 4 \) m

Composite Shapes

Many SSAT problems combine two or more simple shapes. Break the figure apart, find each area, then add (or subtract).

Triangle Rectangle
\(15\text{ m}\)
\(6\text{ m}\)
\(4\text{ m}\)

Regular Polygons

A regular polygon has all sides equal and all angles equal. Its perimeter is simply \( P = ns \) where \( n \) = number of sides and \( s \) = side length.

\(a\)
\(s\)
 Regular Hexagon

Common Mistakes

✕ Using Slant as Height
The height must be perpendicular to the base. Using a slant side gives the wrong area.
✕ Forgetting the \(\tfrac{1}{2}\)
Triangle and trapezoid formulas include \(\tfrac{1}{2}\). Omitting it doubles your answer.
✕ Wrong Units
Perimeter is in linear units (cm). Area is in square units (cm²). Mixing them up loses points.

Formula Summary

Shape Perimeter Area
Rectangle \( 2l + 2w \) \( lw \)
Square \( 4s \) \( s^2 \)
Triangle \( a + b + c \) \( \tfrac{1}{2}bh \)
Parallelogram \( 2a + 2b \) \( bh \)
Trapezoid \( a + b_1 + c + b_2 \) \( \tfrac{1}{2}(b_1 + b_2)h \)
Regular Polygon \( ns \) \( \tfrac{1}{2}(ns)(a) \)

Question 1 of 5

A rectangle has length 15 cm and width 9 cm. What is its area?

Question 2 of 5

A triangle has a base of 14 in and a height of 10 in. What is its area?

Question 3 of 5

A trapezoid has parallel sides of 5 m and 11 m, with a height of 6 m. What is its area?

Question 4 of 5

A square has a perimeter of 36 ft. What is its area?

Question 5 of 5

An L-shaped figure is made of two rectangles: one is \( 8 \text{ m} \times 3 \text{ m} \) and the other is \( 5 \text{ m} \times 3 \text{ m} \). What is the total area?