4.1 Perimeter and Area of Triangles, Rectangles, Polygons & Composite Figures

Why This Topic Matters

GED Testing Service's official Assessment Guide for Educators: Mathematical Reasoning names assessment target Q.4: "Calculate dimensions, perimeter, circumference, and area of two-dimensional figures." Q.4 sits inside the Measurement portion of Quantitative Problem Solving (Quantitative Problem Solving is 45% of the whole test), and it is one of the most reliably tested targets on the exam because two-dimensional shapes show up constantly in real-world contexts GED item writers favor: flooring a room, fencing a yard, tiling a patio, framing a photo. This section covers three of the five Q.4 indicators — Q.4.a ("Compute the area and perimeter of triangles and rectangles. Determine side lengths of triangles and rectangles when given area or perimeter."), Q.4.c ("Compute the perimeter of a polygon. Given a geometric formula, compute the area of a polygon. Determine side lengths of the figure when given the perimeter or area."), and Q.4.d ("Compute perimeter and area of 2-D composite geometric figures, which could include circles, given geometric formulas as needed."). Circles and the Pythagorean theorem (Q.4.b and Q.4.e) get their own section next.

Here is the detail that changes how you should study: the onscreen GED test provides a formula sheet listing every area and perimeter formula you need. That means the test is not measuring whether you memorized "area of a triangle equals one-half base times height" — it is measuring whether you can (1) recognize which shape a word problem describes, (2) plug the right numbers into the right slots of the formula, and (3) rearrange the formula algebraically when the question gives you the area or perimeter and asks for a missing side. That third skill — solving backward — is explicitly named in Q.4.a and Q.4.c and is where most test-takers who "know the formulas" still lose points, because it requires basic equation-solving on top of geometry.

Core Terms and Formulas

Perimeter is the total distance around the outside edge of a two-dimensional figure, measured in linear units (feet, meters, inches). Area is the amount of surface enclosed inside a figure, measured in square units (ft², m², in²). Mixing these two up — giving a linear answer when the question asks for area, or vice versa — is one of the most common wrong-answer traps on the test, and checking whether an answer choice carries a squared unit is a fast way to eliminate distractors.

The GED formula sheet gives you these exact formulas for Q.4.a and Q.4.c:

Two definitions matter more than the formulas themselves. First, in the triangle and parallelogram area formulas, b is the base and h is the height measured perpendicular to that base — not the length of a slanted side. A triangle's slanted side is longer than its true height, so plugging in the slant length instead of the perpendicular height inflates the area. Second, a polygon is any closed figure made of straight sides; for polygons beyond the five listed above (a regular pentagon or hexagon, for instance), Q.4.c specifically tells you the test will give you the geometric formula directly in the item — your job is correct substitution, not recall.

Composite Figures: Split, Then Add or Subtract

A composite figure is a shape built from two or more basic shapes joined together — an L-shaped room, a rectangular patio with a triangular flower bed attached, a picture frame with a rectangular opening cut out of a larger rectangle. Q.4.d tests exactly this. The method is always the same:

1. Draw a line (mentally or on scratch paper) dividing the composite figure into shapes from the formula table.
2. Calculate the area (or perimeter) of each piece separately using the formulas above.
3. Add the pieces if they are joined together to form the total shape; subtract a piece if it represents a hole or cutout removed from a larger shape.

Worked example (addition): A garden is a 12-foot by 8-foot rectangle with a triangular flower bed attached to one 8-foot side. The triangle has a base of 8 feet and a height of 5 feet. Total area = rectangle + triangle = (12 × 8) + (½ × 8 × 5) = 96 + 20 = 116 square feet.

Worked example (subtraction): A rectangular patio measures 20 feet by 15 feet, but a 12-foot by 8-foot rectangular pool is cut out of it and the question asks for the patio area only (not the pool). Patio area = (20 × 15) − (12 × 8) = 300 − 96 = 204 square feet. Adding the pool's area instead of subtracting it is the single most common composite-figure error on this target — always re-read the question to confirm whether the missing piece is included or excluded.

Solving Backward: Given Perimeter or Area, Find a Side

Many Q.4.a and Q.4.c items reverse the direction: they give you the perimeter or area and a relationship between the sides, then ask you to find a specific dimension. This is where geometry and algebra (covered starting in Chapter 8) overlap directly.

Worked example: A rectangular garden has a perimeter of 46 feet. The length is 5 feet more than the width. Find the width.

• Start from the perimeter formula: 2l + 2w = 46, so l + w = 23.
• Substitute the relationship l = w + 5: (w + 5) + w = 23.
• Combine like terms: 2w + 5 = 23, so 2w = 18, and w = 9 feet (length = 14 feet).

Common Traps on Test Day

• Confusing perimeter (add all sides, linear units) with area (multiply dimensions, square units).
• Using a slanted side instead of the true perpendicular height in a triangle or parallelogram.
• Adding a cutout area in a composite figure instead of subtracting it (or vice versa).
• Forgetting that a trapezoid has two parallel bases (b₁ and b₂) that both get added before multiplying by height and dividing by 2.
• Skipping the algebra step when the problem gives a relationship between sides (like "length is 3 less than twice the width") instead of a plain number.

Key Takeaways

• Perimeter = distance around (linear units); area = space enclosed (square units) — never mix the two.
• The GED formula sheet supplies every Q.4.a/Q.4.c formula; your job is identifying the shape, finding perpendicular height, and substituting correctly.
• For composite figures (Q.4.d), split into known shapes and add joined pieces or subtract cutout pieces.
• When a problem gives perimeter/area plus a relationship between sides, substitute that relationship into the formula and solve algebraically for the unknown side.
• Always double-check units in your final answer choice — a square-unit answer to a perimeter question (or vice versa) is a guaranteed wrong choice.