Geometry: choose the right length and area
Distinguish area from perimeter, identify perpendicular height and break composite shapes into simple parts.
In this guide
What quantity is being requested?
Perimeter measures distance around a boundary; area measures a two-dimensional region; volume measures three-dimensional space. Their units differ: metres, square metres and cubic metres. A rectangle of length L and width W has area L × W and perimeter 2L + 2W. Before choosing a formula, decide whether a problem asks about fencing an edge, covering a floor or filling a container. Those verbs are clues, but the actual stated quantity is what determines the calculation.
Worked example: a rectangular floor with an opening
A rectangular floor measures 9 m by 6 m. A rectangular opening of 2 m by 1.5 m occupies part of it. Assume the opening is entirely inside the floor and ask for the remaining floor area. The full rectangle has area 9 × 6 = 54 square metres. The opening has area 2 × 1.5 = 3 square metres. Subtract to obtain 51 square metres. This subtraction answers area only; it does not determine the total exposed boundary unless the arrangement and desired boundary are specified.
A triangle needs a perpendicular height
A triangle with a 14 cm base and perpendicular height of 9 cm has area (1/2) × 14 × 9 = 63 square centimetres. The height is the perpendicular distance from the opposite vertex to the line containing the base. It is not automatically the length of a sloping side. For an obtuse triangle, the foot of the height can lie outside the triangle. A sketch can help you choose the correct quantities, but a drawing that is not stated to be to scale cannot establish a missing measurement.
Use the Pythagorean theorem only for right triangles
A right triangle has legs of 8 m and 15 m. Its hypotenuse c satisfies c^2 = 8^2 + 15^2 = 64 + 225 = 289, so c = 17 m. The hypotenuse is opposite the right angle and is the longest side. If 17 m is known instead and one leg is 8 m, the other leg satisfies b^2 = 17^2 - 8^2 = 225, giving 15 m. Adding all known squares regardless of which side is missing is a common mistake.
Convert units before combining dimensions
A rectangle measuring 80 cm by 1.2 m can be calculated as 0.8 m × 1.2 m = 0.96 square metres, or as 80 cm × 120 cm = 9,600 square centimetres. Both are correct. Because one metre contains 100 centimetres, one square metre contains 100 × 100 = 10,000 square centimetres. Area conversion factors must be squared; using a factor of 100 for square units is insufficient.
- Label each known side and identify any required right angle.
- For a composite area, add nonoverlapping pieces or subtract clearly described holes.
- Do not infer that equal areas imply equal perimeters.
Try it yourself
A right triangle has legs 6 cm and 8 cm. Calculate its area, hypotenuse and perimeter. Then calculate the area and perimeter of a 7 m by 4 m rectangle. Keep the units attached: which answers describe a surface and which describe a length?
Reveal the worked answer
A right triangle has legs 6 cm and 8 cm. Its area is (1/2) × 6 × 8 = 24 square centimetres. Its hypotenuse is the square root of 36 + 64, or 10 cm, so its perimeter is 6 + 8 + 10 = 24 cm. The two answers happen to share the number 24 but have different units and meanings. A 7 m by 4 m rectangle has area 28 square metres and perimeter 22 m; those quantities generally are not numerically equal.
A right triangle has legs 6 cm and 8 cm. Its area is (1/2) × 6 × 8 = 24 square centimetres. Its hypotenuse is the square root of 36 + 64, or 10 cm, so its perimeter is 6 + 8 + 10 = 24 cm. The two answers happen to share the number 24 but have different units and meanings. A 7 m by 4 m rectangle has area 28 square metres and perimeter 22 m; those quantities generally are not numerically equal.
Put one idea into practice.
Try a short mathematics session, then use the explanations to choose what to review next.
Start a short sessionSources & further reading
OpenStax Elementary Algebra 2e — Triangles, rectangles and the Pythagorean theoremBIPM — SI prefixesSomething unclear or incorrect? Suggest a correction to this guide.
Original FreeReviewer.ph guidance, informed by the linked sources. Exam requirements and procedures should always be checked with the examining body.