Area and Perimeter of Shapes

Length units and square units

Perimeter is a length, so it uses units such as centimetres or metres. Area uses square units, such as square centimetres, written $\mathrm{cm}^2$. A square centimetre is the area of a square whose sides are each 1 cm.

For a rectangle measuring 6 cm by 4 cm, the perimeter is the total of its four side lengths:

Its area is the number of unit squares that cover it:

The numbers 20 and 24 describe different quantities. Neither is the “larger answer” to the same measurement question.

Conversions must account for both dimensions. Since $1\text{ m}=100\text{ cm}$,

Thus $0.75\text{ m}^2=7\,500\text{ cm}^2$, not 75 square centimetres. Convert mixed length units before multiplying. A panel 2.4 m long and 80 cm wide has width 0.8 m, so its area is $2.4\times0.8=1.92\text{ m}^2$.

Rectangles and squares

A rectangle with length $l$ and width $w$ has two sides of each length. Therefore

A square has $l=w=s$, which gives

These formulas encode the structure of the shape. The perimeter formula adds boundary lengths. The area formula multiplies two perpendicular dimensions. See OpenStax’s rectangle and triangle treatment for a reference derivation.

Suppose a rectangular garden is 8 m long and 5 m wide. Its area is $40\text{ m}^2$ and its perimeter is 26 m. If a 1.2 m gate replaces one section of fencing, the remaining fence length is $26-1.2=24.8\text{ m}$. The gate changes the amount of fencing needed, not the garden’s area.

Same perimeter does not mean same area

A 6 cm by 4 cm rectangle and an 8 cm by 2 cm rectangle both have perimeter 20 cm. Their areas are 24 and 16 square centimetres respectively. Boundary length alone does not determine the area of a general rectangle.

Likewise, regions can share an area without sharing a perimeter. Always calculate the requested quantity from the available dimensions rather than treating one measurement as a substitute for the other.

Parallelograms: use perpendicular height

For a parallelogram, choose a base $b$ and measure the perpendicular distance $h$ between the parallel lines containing that base and the opposite side. Its area is

The height is not generally the slanted side. Cutting a triangular piece from one end and moving it to the other forms a rectangle with the same base and height. Rearrangement changes the outline but preserves area.

In the model, the base is 8 cm, the perpendicular height is 3 cm, and the slanted side is 5 cm. The area is $8\times3=24\text{ cm}^2$. The perimeter uses the slanted boundary side instead:

Using 5 as the height would calculate a different rectangle’s area, not the area of this parallelogram.

Triangles: half a parallelogram

Two congruent copies of a triangle can form a parallelogram. Its area is therefore

Here $h$ must be perpendicular to the chosen base or its extension. For an obtuse triangle, the perpendicular height can lie outside the triangle. That does not change the formula: height measures separation from the line containing the base.

A right triangle with perpendicular legs 6 cm and 8 cm has area

To find its perimeter, calculate the third side using the Pythagorean theorem:

The numerical answers both happen to be 24, but one is an area and the other is a length. This coincidence does not make their formulas interchangeable.

For a non-right triangle, do not multiply two arbitrary side lengths and divide by two. You need a base and its matching perpendicular height, or a different formula supported by the information provided.

Trapezoids: average the parallel bases

For a trapezoid with parallel bases of lengths $a$ and $b$ and perpendicular height $h$,

Two congruent copies form a parallelogram whose base is $a+b$ and whose height is $h$. Dividing that area by two gives the formula. Equivalently, the area is the average base length multiplied by the height.

If the parallel bases measure 8 cm and 14 cm, and the height is 4 cm,

The perimeter requires the other two side lengths as well. The bases and height alone do not always determine them. If the trapezoid is also isosceles and centred, the 6 cm difference between its bases splits into two 3 cm offsets. Each slanted leg is then $\sqrt{3^2+4^2}=5$ cm, so its perimeter is $8+14+5+5=32$ cm.

Notice the additional condition. The area calculation did not need an isosceles trapezoid; that condition was used to determine its missing boundary lengths.

Circles: circumference and enclosed area

For radius $r$ and diameter $d=2r$, a circle’s boundary length, called circumference, is

Its enclosed area is

The constant $\pi$ is the ratio of circumference to diameter. It is not a unit. OpenStax reviews circles and composite regions.

For a circular mat of radius 3 m, the exact circumference is $6\pi$ m and the exact area is $9\pi\text{ m}^2$. Rounded to two decimal places, these are approximately 18.85 m and 28.27 square metres. Keep $\pi$ in the intermediate steps and round the final measurement, rather than repeatedly replacing it with 3.14.

A semicircle’s area is half the full disk area. Its complete perimeter includes the straight diameter:

Half a circumference measures only the curved part, not the entire boundary.

Sectors and arc length

A sector with central angle $\theta$, measured in degrees, occupies the fraction $\theta/360^\circ$ of a disk. For a non-degenerate sector smaller than a full disk,

Its perimeter is the arc length plus two radii. A sector with radius 6 cm and central angle $60^\circ$ has area $6\pi\text{ cm}^2$, arc length $2\pi$ cm and perimeter $(12+2\pi)$ cm. The two radii are distinct boundary segments here. For a full disk, there are no radial boundary edges to add.

Composite shapes: add areas, trace boundaries

A composite region can often be split into rectangles, triangles or sectors whose areas are known. You can also start with a larger region and subtract a cut-out. The pieces must not overlap in area, and their dimensions must fit the stated shape.

Begin with a 10 m by 8 m rectangle and remove a 4 m by 3 m rectangle from its upper-right corner. The remaining L-shaped area is

For perimeter, trace the remaining boundary once. The side lengths are 10, 5, 4, 3, 6 and 8 metres, giving

The original rectangle also had perimeter 36 m. Removing this corner replaced two outer sections with two inner sections of equal combined length. This is a feature of this rectangular corner cut, not a rule that removing material never changes perimeter.

A hole changes which boundaries count

Now remove a 4 m by 3 m rectangle completely inside the same 10 m by 8 m rectangle. The area is still $68\text{ m}^2$. The outer boundary remains 36 m, but the hole adds an inner boundary of $2(4+3)=14$ m. The total boundary length of the remaining region is 50 m.

A question about outside fencing requires 36 m. A question about edging every exposed boundary requires 50 m. Specify the boundary being measured. Internal dividing lines used to calculate area are not extra perimeter unless they are actual edges of the region.

Work backwards from an area

Formulas can determine a missing measurement. If a triangle has area $42\text{ cm}^2$ and base 12 cm,

If a rectangle has area $54\text{ cm}^2$ and width 6 cm, its length is $54/6=9$ cm and its perimeter is $2(9+6)=30$ cm.

Write the formula before rearranging it, and identify the units of the unknown. Dividing an area by a length gives a length. A result in square centimetres for a missing height signals a units error even if the arithmetic is correct.

Scale changes length and area differently

When all corresponding lengths are multiplied by the same positive scale factor $k$, perimeter is multiplied by $k$, while area is multiplied by $k^2$:

Doubling a rectangle’s length and width doubles its perimeter and quadruples its area. If only one dimension changes, the shape has not been uniformly scaled and you should recalculate rather than apply the similarity rule blindly.

On a 1:50 plan, a room shown as 8 cm by 6 cm measures 4 m by 3 m in reality. Its actual area is 12 square metres. The plan area is $48\text{ cm}^2$, and the real area in square centimetres is $48\times50^2=120\,000\text{ cm}^2$. Both routes give the same result after conversion.

Formula reference

Use this reference after identifying the shape, the requested quantity and the dimensions. Identify the quantity represented by each symbol before substituting measurements.

Related topics

Review Geometric Shapes in Maths when the shape classification is uncertain. Continue with Surface Areas and Volumes to extend square-unit reasoning to the surfaces and interiors of solids.