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GED Math: Perimeter, Area, and the Pythagorean Theorem Concepts

Importance ★★★★☆ · 20% (estimate)

Overview

This chapter covers flat shapes: the perimeter (distance around) and area (space inside) of rectangles, squares, triangles, parallelograms, trapezoids, circles, and shapes made by combining them. It also covers the Pythagorean theorem for right triangles. The GED formula sheet lists these formulas, so the main job is choosing the right one and putting in the right numbers.

Perimeter is measured in plain units such as feet, while area is measured in square units such as square feet. A triangle's area is half of base times height. For a circle, the radius is the distance from the center to the edge and the diameter is twice the radius. The circumference is the distance around the circle, and the test uses 3.14 for pi.

For a composite shape, split it into simple shapes, find each area, and add them, or subtract a piece that has been cut out. When an area or perimeter is given, you can work backward to find a missing side or radius.

In a right triangle, the two shorter sides (legs) and the longest side (hypotenuse) are linked by a² + b² = c². This is useful for ladders, ramps, diagonals of rectangles, and TV screen sizes.

Key concepts

  • Rectangle and square

    Rectangle: A = lw and P = 2l + 2w. Square: A = s² and P = 4s.

    Memory tipArea is inside (square units); perimeter is around (plain units).

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Triangle, parallelogram, trapezoid

    Triangle: A = 1/2 bh. Parallelogram: A = bh. Trapezoid: A = 1/2 h(b₁ + b₂). The height is always measured straight up from the base.

    Memory tipDo not forget the 1/2 for triangles and trapezoids.

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Circle

    Area A = πr². Circumference C = 2πr or C = πd. Use π ≈ 3.14. The diameter is twice the radius.

    Memory tipCheck whether the problem gives the radius or the diameter.

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Pythagorean theorem

    In a right triangle, a² + b² = c², where c is the longest side, across from the right angle.

    Memory tipTo find a shorter side, subtract: b² = c² − a².

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Composite figures

    A shape made of simpler shapes. Find each area and add them, or subtract the area of a cut-out part.

    Memory tipA half circle has half the area of the full circle with the same radius.

Often tested

  • Working backward: r = √(A ÷ π) and d = C ÷ π.
  • Common right triangles: 3-4-5, 5-12-13, and their multiples such as 6-8-10.
  • The hypotenuse is always the longest side, so it must be longer than either leg.
  • When a perimeter is given with a relationship between sides, write an equation for the width first.

Easy to confuse: Area and circumference of a circle

ItemAreaCircumference
Formulaπr²2πr or πd
MeasuresSpace insideDistance around
UnitsSquare unitsPlain units
Radius 5 (π = 3.14)78.531.4

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 1. A rectangular room is 12 feet wide and 15 feet long. How many square feet of carpet are needed to cover the floor?

  1. ① 180
  2. ② 54
  3. ③ 360
  4. ④ 27
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ① 180

Key point: Rectangle area A = l × w

  1. Area of a rectangle: A = length × width.
  2. A = 15 × 12 = 180 square feet.

Check: 10 × 15 = 150 and 2 × 15 = 30. 150 + 30 = 180.

Wrong choices

  • 54: 54 is the perimeter (2 × 12 + 2 × 15), the distance around the room.
  • 27: 27 adds the length and width instead of multiplying.
  • 360: 360 doubles the area; there is no need to multiply by 2.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 2. The bottom of a ladder is 5 feet from a wall. The top of the ladder touches the wall 12 feet above the ground. How long is the ladder?

  1. ① 7 feet
  2. ② 13 feet
  3. ③ 169 feet
  4. ④ 17 feet
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Answer: ② 13 feet

Key point: Right triangle: a² + b² = c²

  1. The wall, the ground, and the ladder form a right triangle. The ladder is the longest side, c.
  2. a² + b² = c²: 5² + 12² = 25 + 144 = 169.
  3. c = √169 = 13 feet.

Check: 13² = 169 = 25 + 144.

Wrong choices

  • 17 feet: 17 feet adds 5 and 12; the sides must be squared first.
  • 169 feet: 169 is c², not c. Take the square root.
  • 7 feet: 7 feet subtracts 5 from 12; the ladder must be longer than either side.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 3. A rectangular garden is 18 feet long and 10 feet wide. How many feet of fencing are needed to go all the way around it?

  1. ① 180 feet
  2. ② 46 feet
  3. ③ 56 feet
  4. ④ 28 feet
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ③ 56 feet

Key point: Rectangle perimeter P = 2l + 2w

  1. Perimeter: P = 2l + 2w.
  2. P = 2 × 18 + 2 × 10 = 36 + 20 = 56 feet.

Check: Walk around the garden: 18 + 10 + 18 + 10 = 56 feet.

Wrong choices

  • 28 feet: 28 feet covers only one length and one width, which is half the way around.
  • 180 feet: 180 is the area in square feet, not the distance around.
  • 46 feet: 46 feet leaves out one of the 10-foot sides.
Practice 5 questions from this unit →

AI-written · Verified · Written by AI. Only units that passed a separate AI check (re-reading facts and numbers) are published. Tell us if you find a mistake.