Perimeter and area of polygons

Elvin's family wants to lay laminate flooring in a 5 m by 4 m room and fit skirting boards along the bottom of the walls. The amount of laminate depends on the room's area, and the length of skirting on its perimeter. Mix the two up and you either waste money or run back to the shop halfway through the job. In this lesson you will learn to tell perimeter and area apart, convert units of area, and use the area formulas of the main polygons — not by memorising them, but by seeing where each one comes from.

Perimeter

You need no special formula for a perimeter — adding up all the sides is enough. But when sides are equal, the adding turns into multiplying and gives short formulas. A rectangle has two lengths and two widths, a square has four equal sides, and a regular polygon (one whose sides and angles are all equal) with n sides has n equal sides.

Area and units of area

Every area formula rests on two simple properties of area. First, congruent figures have equal areas: if you cut a figure up and rearrange the pieces, the area does not change. Second, if a figure is split into several parts, its area is the sum of the areas of the parts. Below we derive every formula from the rectangle using exactly these two properties.

When the unit of length grows 10 times, both sides of the unit square grow 10 times, so the unit of area grows 10 · 10 = 100 times. For example, 1 dm = 10 cm, so 1 dm² holds 10 rows of 10 squares of 1 cm² each, that is 100 of them. Land has special units too: an are is the area of a square with 10 m sides, and a hectare is the area of a square with 100 m sides.

Rectangle, square and parallelogram

Cover a 5 cm by 3 cm rectangle with 1 cm² squares: 5 squares in each row, 3 rows — 5 · 3 = 15 squares. So the area is 15 cm². The same reasoning works for any sides (for fractional ones too: then we count smaller squares, for example mm²). A square is a rectangle with equal sides, so its area is a · a = a². That is also why the second power of a number is called its “square”.

Perimeter and area do not determine each other. Murad has 20 m of fencing and wants to fence off a rectangular vegetable patch. In every option the perimeter is 20 m, yet the area changes a lot:

Now the parallelogram. Drop the height BH from vertex B to the base AD. It cuts off the right triangle ABH. Move this triangle to the other end, next to side DC, and you get the rectangle HBCK with sides a and h. Nothing was lost and nothing was added, so the areas are equal: A = a · h. The height of a parallelogram is the distance between two opposite sides; the slanted side does not affect the area.

Triangle, rhombus and trapezoid

Take a congruent copy of any triangle ABC, turn it through 180° and attach it along side BC. Together the two congruent triangles form the parallelogram ABDC with base a and height h. The area of the parallelogram is a · h, and the triangle is exactly half of it. In a right triangle this is even easier to see: it is half of a rectangle whose sides equal the legs, cut along its diagonal.

As you know from the lesson “Quadrilaterals: parallelogram, rectangle, rhombus, square and trapezoid”, the diagonals of a rhombus are perpendicular and bisect each other. Draw lines through the vertices of the rhombus parallel to the diagonals: you get a rectangle with sides d₁ and d₂. The diagonals split it into 8 congruent right triangles, and 4 of them lie inside the rhombus. So the rhombus is exactly half of that rectangle: A = ½ · d₁ · d₂. A rhombus is also a parallelogram, so A = a · h holds as well.

Split the trapezoid by the diagonal AC into two triangles. Triangle ACD has base AD = a, and triangle ABC has base BC = b. Their heights are the same — the height h of the trapezoid, because the bases are parallel. Add the areas: A = ½ · a · h + ½ · b · h = (a + b)/2 · h. The expression (a + b)/2 is the midline of the trapezoid, so in short A = m · h.

Composite shapes and home-repair problems

Rooms, plots of land and machine parts are rarely simple shapes. Then we use the second property of area: we split the shape into familiar pieces, or complete it to a large rectangle and subtract the extra part. Both ways must give the same answer — which is also a good check.

1. 1
   1. Draw it and label the lengths

   Mark the length of every side on the drawing. Find missing sides as the difference or the sum of opposite sides.

2. 2
   2. Split or complete

   Split the shape into rectangles, triangles and trapezoids, or draw the rectangle around it and mark the pieces that are missing.

3. 3
   3. Find the area of each piece

   Apply the right formula; all lengths must be in the same unit.

4. 4
   4. Add or subtract

   When splitting, add the areas; when completing, subtract the missing pieces from the area of the large rectangle.

5. 5
   5. Check

   Check the answer by the other method and make sure the unit is a square unit. Count the perimeter separately, along the boundary.

In this lesson we derived every area formula from a single rectangle. In the lesson “The Pythagorean theorem” you will see the famous link between the areas of the squares built on the sides of a right triangle, and circles are studied in the lesson “Circumference and area of a circle” — where the area of a disc also comes from the parallelogram formula.