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GED · Mathematical Reasoning · 6 min read

Perimeter, Area, Circles, and Angles

In this lesson you will
  • Calculate the perimeter and area of basic polygons and composite figures using standard geometric formulas.
  • Solve problems involving circles and semicircles, including finding circumference, area, radius, and diameter.
  • Identify angle relationships and solve for unknown angle measurements in triangles, quadrilaterals, and parallel lines cut by a transversal.

Perimeter of Polygons and L-Shaped Figures

Perimeter is the total distance around the outside boundary of a two-dimensional shape. It represents the path you would walk if you traveled completely around the edges of a flat figure. Perimeter is the distance around a shape’s boundary, which may include straight or curved parts, and is measured in linear units such as inches, feet, or centimeters.

Specific geometric shapes have direct formulas for finding perimeter. For a rectangle, the perimeter is found using the formula P = 2l + 2w, where l is the length and w is the width. For a square, because all four sides are the same length, the perimeter is calculated using P = 4s, where s is the side length. If you are working with an irregular polygon where the perimeter and all side lengths except one are known, you can find the missing side length by subtracting the sum of the known sides from the total perimeter.

Some figures have an L-shaped outline made up entirely of right angles. To find missing sides in a closed L-shaped figure, look at the opposite directions. The total distance along horizontal sides going left equals the total distance along horizontal sides going right. Similarly, the total distance along vertical sides going up equals the total distance along vertical sides going down. You can use these two balancing relationships to solve for any missing vertical or horizontal lengths before calculating the total perimeter.

Exam tip. When finding the perimeter of an L-shaped figure, test questions often leave two sides unlabeled. Do not guess their measurements. Use the rule that the outer vertical side equals the sum of the opposing vertical segments, and the outer horizontal side equals the sum of the opposing horizontal segments, to find all missing numbers before adding.

Common mistakes

  • Using a slanted side instead of the perpendicular height when calculating the area of a triangle or parallelogram.
  • Plugging the diameter directly into the area formula A = πr² instead of dividing the diameter by two to get the radius first.
  • Forgetting to add the straight diameter edge when calculating the perimeter of a closed semicircle (using only πr instead of πr + d).
  • Confusing complementary angles (sum to 90°) with supplementary angles (sum to 180°).

Key points

  • Perimeter measures the distance around a shape in linear units (P = 2l + 2w for rectangles, P = 4s for squares), while area measures the internal space in square units.
  • In an L-shaped figure with right angles, opposing horizontal distances balance each other, and opposing vertical distances balance each other.
  • Area formulas include A = lw (rectangle), A = s² (square), A = bh (parallelogram), A = 1/2 bh (triangle), and A = 1/2 h(b1 + b2) (trapezoid), where height must always be perpendicular to the base.
  • Circle formulas use the radius (half the diameter): circumference is C = 2πr or C = πd, and area is A = πr² (with standard formulas provided on the GED mathematics formula sheet).
  • A closed semicircle has an area of 1/2 πr² and a perimeter of πr + d.
  • Interior angles sum to 180° in any triangle and 360° in any quadrilateral, with right triangle acute angles adding to 90°.
  • When a transversal cuts two parallel lines, corresponding, alternate interior, and alternate exterior angles are equal, while consecutive interior angles sum to 180°.

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Original lesson written for GED practice · updated October 30, 2026. Exam names are trademarks of their owners; tests.global is not affiliated with them.