Geometria

Area and Perimeter of Polygons: Solved Exercises From Easy to Hard

The difference between area and perimeter, the formulas for the most common polygons, and five solved exercises of increasing difficulty with step-by-step verified answers.

Recommended for: Grade 6 · Grade 7 · Grade 8

The perimeter of a polygon is the sum of the lengths of all its sides, measured in cm; the area is the size of the surface enclosed by the polygon, measured in cm². They are two different quantities: the first is about the border, the second about what sits inside.

The perimeter always works the same way

For any polygon the perimeter is found the same way: add up every side. The short formulas only exist to save time when some sides are equal. In a square P = 4 × s, in a rhombus P = 4 × s, in a rectangle P = 2 × (b + h). If no sides are equal, you just add.

The area formulas you actually need

Area, on the other hand, has one rule per shape:

  • rectangle: A = b × h
  • square: A = s × s
  • triangle: A = (b × h) / 2
  • parallelogram: A = b × h
  • rhombus: A = (d₁ × d₂) / 2
  • trapezoid: A = ((B + b) × h) / 2

The most common mistake is not forgetting a formula, it is using a slanted side as the height. The height is always perpendicular to the base.

When the data is not enough

Many exercises do not hand you the number you need. If you are asked for the perimeter of a rhombus and given the diagonals, you get the side from the Pythagorean theorem. If you are asked for the perimeter of a right trapezoid, the slanted side comes out the same way. So before calculating, always ask: which measurement am I missing, and where can I get it from?

Perimeter and area do not move together

Two shapes with the same perimeter can have very different areas, and you can cut a piece out of a shape so that its area drops while its perimeter stays the same. The last exercise below shows exactly that case.

The exercises that follow use the same pattern at increasing difficulty: they start with a rectangle where the data is ready to use, move on to a rhombus and a trapezoid where one measurement has to be worked out, and end with a composite shape. Every answer was verified by redoing the calculations.

Solved exercises

1. A rectangle has a base of 12 cm and a height of 7 cm. Find its perimeter and area. base

Show solution
  1. The perimeter is the sum of the four sides; in a rectangle the sides are equal in pairs, so P = 2 × (b + h).
  2. P = 2 × (12 + 7) = 2 × 19 = 38 cm.
  3. The area of a rectangle is A = b × h.
  4. A = 12 × 7 = 84 cm².

Answer: P = 38 cm ; A = 84 cm²

2. A right triangle has legs of 6 cm and 8 cm. Find its perimeter and area. base

Show solution
  1. For the perimeter I also need the hypotenuse, which I get from the Pythagorean theorem.
  2. h = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
  3. P = 6 + 8 + 10 = 24 cm.
  4. In a right triangle the two legs act as base and height: A = (6 × 8) / 2 = 48 / 2 = 24 cm².
  5. Careful: the numbers come out the same, but 24 cm and 24 cm² are different measurements.

Answer: P = 24 cm ; A = 24 cm²

3. A rhombus has diagonals of 12 cm and 16 cm. Find its area and perimeter. intermedio

Show solution
  1. The area of a rhombus comes from its diagonals: A = (d₁ × d₂) / 2.
  2. A = (12 × 16) / 2 = 192 / 2 = 96 cm².
  3. For the perimeter I need the side. The diagonals of a rhombus bisect each other at right angles, forming four right triangles with legs 12/2 = 6 cm and 16/2 = 8 cm.
  4. side = √(6² + 8²) = √100 = 10 cm.
  5. A rhombus has four equal sides: P = 4 × 10 = 40 cm.

Answer: A = 96 cm² ; P = 40 cm

4. A right trapezoid has a longer base of 14 cm, a shorter base of 8 cm and a height of 8 cm. Find its area and perimeter. intermedio

Show solution
  1. Area of a trapezoid: A = ((B + b) × h) / 2.
  2. A = ((14 + 8) × 8) / 2 = (22 × 8) / 2 = 176 / 2 = 88 cm².
  3. For the perimeter I am missing the slanted side. In a right trapezoid one side equals the height (8 cm), while the slanted side is the hypotenuse of a right triangle whose legs are the height and the difference of the bases.
  4. difference of the bases = 14 − 8 = 6 cm ; slanted side = √(8² + 6²) = √(64 + 36) = √100 = 10 cm.
  5. P = 14 + 8 + 8 + 10 = 40 cm.

Answer: A = 88 cm² ; P = 40 cm

5. A 5 cm square is cut out of one corner of a 20 cm × 12 cm rectangle, leaving an L-shaped figure. Find the area and the perimeter of the L shape. avanzato

Show solution
  1. Area of the original rectangle: 20 × 12 = 240 cm².
  2. Area of the square removed: 5 × 5 = 25 cm².
  3. Area of the L shape: 240 − 25 = 215 cm².
  4. For the perimeter I walk around the outline: 20 + 7 + 5 + 5 + 15 + 12.
  5. P = 64 cm. It is the same perimeter as the original rectangle, because the two sides of the square taken off the outline (5 + 5) are replaced exactly by the two sides of the cut (5 + 5).
  6. Conclusion: the area dropped by 25 cm², the perimeter did not change.

Answer: A = 215 cm² ; P = 64 cm

FAQ

What is the difference between area and perimeter?

The perimeter measures the outline of a shape, that is how long the line around it is; the area measures the surface inside, that is how much space it covers. Fencing a garden is a perimeter problem, laying the lawn is an area problem.

Why is area measured in cm² and perimeter in cm?

The perimeter is a sum of lengths, so it stays a length (cm). The area comes from multiplying two lengths, and cm times cm gives square centimetres (cm²).

Do two shapes with the same perimeter always have the same area?

No. A 5 cm × 1 cm rectangle and a square with 3 cm sides both have a perimeter of 12 cm, but the first has an area of 5 cm² and the second 9 cm². For a given perimeter, the more compact the shape, the larger its area.