Area and Perimeter

The One Thing

Perimeter measures the total length of the outside boundary of a closed plane figure, whereas area measures the enclosed region. Solve problems by identifying the shape or component shapes, selecting the appropriate formula, using compatible units and perpendicular heights, and including only the required boundary.

Definitions and Results

• Perimeter: The total length of the boundary of a closed plane figure, measured in linear units such as cm or m.
• Area: The region enclosed by a plane figure, measured in square units such as cm² or m².
• Rectangle: A quadrilateral with four right angles.

- Perimeter: $2(l+b)$ - Area: $l\times b$

• Square: A quadrilateral with four equal sides and four right angles.

- Perimeter: $4a$ - Area: $a^2$

• Triangle: A polygon with three sides.

- Area: $\frac{1}{2}\times \text{base}\times \text{corresponding height}$ - Perimeter: ${\text{side}}_1+{\text{side}}_2+{\text{side}}_3$

• Parallelogram: A quadrilateral with both pairs of opposite sides parallel.

- Area: $\text{base}\times \text{corresponding height}$ - Perimeter: $2(a+b)$

• Rhombus: A parallelogram with all four sides equal.

- Area: $\frac{1}{2}{d}_1{d}_2$, where $d_1$ and $d_2$ are the diagonals - Perimeter: $4a$

• Trapezium: A quadrilateral with one pair of parallel sides.

- Area: $\frac{1}{2}(a+b)h$, where $a$ and $b$ are the parallel sides and $h$ is the perpendicular height

• Circle: A plane figure whose points are at a fixed distance from a centre.

- Circumference: $2\pi r=\pi d$ - Area: $\pi r^2$

• Radius and diameter: The radius is the distance from the centre to the circle. The diameter is twice the radius:

$$d=2r$$

• Composite figure: A figure formed by combining two or more simple shapes, such as rectangles, triangles, semicircles or circles.
• Decomposition: Dividing a complex figure into familiar shapes so that their areas or boundary lengths can be calculated.
• Unit conversion: Measurements must be converted into compatible units before calculation. For example:

$$1\text{ m}=100\text{ cm},1{\text{ m}}^2=10,000{\text{ cm}}^2$$

• Perpendicular height: In area formulas, the height is the perpendicular distance to the chosen base, not necessarily a slanting side.
• Circle approximation: Use $\pi =\frac{22}{7}$ or $3.14$ when a value is not specified.
• Scale factor: When a figure is enlarged, its perimeter changes by the scale factor, while its area changes by the square of the scale factor.
• Pythagorean theorem: A diagonal of a rectangle or square may be found using

$${\text{hypotenuse}}^2={\text{base}}^2+{\text{height}}^2.$$

• Units: Perimeter is expressed in linear units, whereas area is expressed in square units.

Worked Methods

Finding the perimeter and area of a rectangle

• Identify the length $l$ and breadth $b$.
• Use the perimeter formula:

$$P=2(l+b).$$

• Use the area formula:

$$A=l\times b.$$

• State the perimeter in linear units and the area in square units.

Finding the perimeter and area of a square

• Identify the side length $a$.
• Calculate the perimeter:

$$P=4a.$$

• Calculate the area:

$$A=a^2.$$

Finding the perimeter and area of a triangle

• Identify the three side lengths.
• Add the three sides to obtain the perimeter:

$$P={\text{side}}_1+{\text{side}}_2+{\text{side}}_3.$$

• Identify the chosen base and its corresponding perpendicular height.
• Calculate the area:

$$A=\frac{1}{2}\times \text{base}\times \text{height}.$$

Finding the perimeter and area of a parallelogram

• Identify the two distinct side lengths $a$ and $b$.
• Calculate the perimeter:

$$P=2(a+b).$$

• Identify the base and its corresponding perpendicular height.
• Calculate the area:

$$A=\text{base}\times \text{height}.$$

Finding the perimeter and area of a rhombus

• Identify the common side length $a$.
• Calculate the perimeter:

$$P=4a.$$

• Identify the diagonals $d_1$ and $d_2$.
• Calculate the area:

$$A=\frac{1}{2}{d}_1{d}_2.$$

Finding the area of a trapezium

• Identify the parallel sides $a$ and $b$.
• Identify the perpendicular height $h$.
• Add the parallel sides.
• Multiply by the height and divide by $2$:

$$A=\frac{1}{2}(a+b)h.$$

Finding the circumference and area of a circle

• Identify whether the radius $r$ or diameter $d$ is given.
• If the diameter is given, find the radius using:

$$r=\frac{d}{2}.$$

• Calculate the circumference using either:

$$C=2\pi r\text{or}C=\pi d.$$

• Calculate the area:

$$A=\pi r^2.$$

• Use $\pi =\frac{22}{7}$ or $3.14$ when no value is specified.

Solving a composite figure by decomposition

• Label the diagram and identify all known measurements.
• Divide the composite figure into familiar shapes, such as rectangles, triangles, semicircles or circles.
• Calculate the area of each component using the appropriate formula.
• Add the component areas to obtain the total area.
• For the perimeter, add only the lengths on the outside boundary. Do not include internal dividing lines unless an internal boundary is specifically required.
• Check the result by using an equivalent decomposition where possible; equivalent decompositions of the same figure must produce the same total area.

Solving a composite figure by subtraction

• Enclose the figure within a larger known shape.
• Calculate the area of the larger shape.
• Calculate the area of each missing or unwanted part.
• Subtract the unwanted areas from the larger area:

$$\text{required area}=\text{larger area}-\text{missing area}.$$

• Check that the resulting area is positive and reasonable.

Converting units before calculation

• Identify the units of every given measurement.
• Convert all lengths into one compatible linear unit before calculating perimeter or area.
• Remember that square units require squared conversion factors:

$$1\text{ m}=100\text{ cm},1{\text{ m}}^2=10,000{\text{ cm}}^2.$$

• Give the final answer in linear units for perimeter and square units for area.

Finding a diagonal using the Pythagorean theorem

• Identify the base and height of the rectangle or square.
• Treat the diagonal as the hypotenuse of a right-angled triangle.
• Substitute into:

$${\text{hypotenuse}}^2={\text{base}}^2+{\text{height}}^2.$$

• Take the square root to obtain the diagonal.

Checking an area or perimeter answer

• Confirm that the correct formula was selected for each shape.
• Confirm that the height used in an area formula is perpendicular to the base.
• Check that all units were made compatible before substitution.
• Verify that only the required outer boundary was included for perimeter.
• Check that the final units are appropriate and that the answer is reasonable.

Where It Goes Wrong

• Confusing perimeter with area: perimeter measures a boundary in linear units, while area measures an enclosed region in square units.
• Using a slanting side instead of the perpendicular height in the area formula for a triangle, parallelogram or trapezium.
• Including internal dividing lines when calculating the outside perimeter of a composite figure.
• Failing to convert measurements into compatible units, particularly when converting between ${\text{m}}^2$ and ${\text{cm}}^2$.
• Using the diameter as the radius in the circle area formula, despite the condition $d=2r$.
• Giving an answer without checking the diagram, the required boundary or region, the formula, and the appropriate units.

What Gets Asked

This material supports questions requiring students to:

• Define perimeter, area, composite figure, decomposition, radius and diameter.
• Calculate the perimeter and area of rectangles, squares, triangles, parallelograms, rhombi and trapezia.
• Calculate the circumference and area of circles using $2\pi r$, $\pi d$ and $\pi r^2$.
• Convert between linear units and square units, including $1\text{ m}=100\text{ cm}$ and $1{\text{ m}}^2=10,000{\text{ cm}}^2$.
• Find missing dimensions, including a circle’s radius from its diameter.
• Find a rectangle or square diagonal using the Pythagorean theorem.
• Decompose composite figures into rectangles, triangles, semicircles or circles and calculate their total area.
• Find the area of a composite figure by subtracting missing parts from a larger shape.
• Determine the outside perimeter of a composite figure without counting internal boundaries.
• Apply a scale factor to determine how perimeter and area change.
• Explain why equivalent decompositions of the same figure should produce the same total area.
• Present a complete solution with a labelled diagram, known measurements, formula, substitution, calculation, appropriate units and a reasonableness check.