Perimeter and Area of Plane Shapes

Introduction to Perimeter and Area of Plane Shapes

Perimeter and area are two fundamental measurements used to describe the size of two-dimensional (flat) shapes. The perimeter is the total distance around the outside boundary of a shape, while the area is the amount of surface or space enclosed within the shape. Both concepts are widely used in everyday life, such as in construction, farming, interior design, and land measurement, making them essential mathematical skills.

Meaning and Units of Perimeter

The perimeter of a shape is calculated by adding together the lengths of all its sides. Perimeter is measured in units of length, such as centimetres (cm), metres (m), or kilometres (km). For example, if a rectangular garden has a length of 15 m and a width of 8 m, its perimeter is 15 + 8 + 15 + 8 = 46 m, representing the total distance around the garden's boundary, useful for calculating how much fencing material would be needed.

Perimeter of Common Shapes

The perimeter of a rectangle is calculated as P = 2(length + width) or P = 2l + 2w. The perimeter of a square, where all four sides are equal, is P = 4 × side length. The perimeter of a triangle is found by adding the lengths of all three sides together: P = a + b + c. The perimeter of any polygon (a shape with straight sides) is found by simply adding together the lengths of all its sides.

Circumference of a Circle

The perimeter of a circle is given a special name: the circumference. The circumference is calculated using the formula C = 2πr (or equivalently, C = πd), where r is the radius (the distance from the centre to the edge of the circle), d is the diameter (the distance across the circle through its centre, equal to twice the radius), and π (pi) is a special mathematical constant approximately equal to 3.14159, often rounded to 22/7 or 3.14 for calculations. For example, a circle with a radius of 7 cm has a circumference of C = 2 × (22/7) × 7 = 44 cm.

Meaning and Units of Area

The area of a shape measures the amount of two-dimensional space it covers, and is measured in square units, such as square centimetres (cm²), square metres (m²), or hectares (for larger areas of land). Area answers questions such as how much paint is needed to cover a wall, how much carpet is needed for a floor, or how much land is available for farming.

Area of a Rectangle and Square

The area of a rectangle is calculated as A = length × width (A = l × w). For example, a rectangular room measuring 6 m by 4 m has an area of 6 × 4 = 24 m². The area of a square, since all sides are equal, is A = side × side (A = s²). For example, a square plot of land with sides of 10 m has an area of 10 × 10 = 100 m².

Area of a Triangle

The area of a triangle is calculated as A = ½ × base × height, where the base is any one side of the triangle, and the height is the perpendicular (right-angle) distance from that base to the opposite vertex (corner). For example, a triangle with a base of 12 cm and a height of 8 cm has an area of A = ½ × 12 × 8 = 48 cm². It is important to use the perpendicular height, not the length of a slanted side, when calculating the area.

Area of a Parallelogram

The area of a parallelogram (a four-sided shape with two pairs of parallel sides) is calculated as A = base × height, using the perpendicular height between the two parallel sides, similar to the triangle formula but without the factor of one-half. For example, a parallelogram with a base of 10 cm and a height of 6 cm has an area of A = 10 × 6 = 60 cm².

Area of a Trapezium

The area of a trapezium (a four-sided shape with exactly one pair of parallel sides) is calculated as A = ½ × (sum of parallel sides) × height, or A = ½(a + b)h, where a and b are the lengths of the two parallel sides, and h is the perpendicular distance between them. For example, a trapezium with parallel sides of 8 cm and 12 cm, and a height of 5 cm, has an area of A = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50 cm².

Area of a Circle

The area of a circle is calculated using the formula A = πr², where r is the radius of the circle. For example, a circle with a radius of 7 cm has an area of A = (22/7) × 7² = (22/7) × 49 = 154 cm². Note that unlike the circumference formula, the radius is squared in the area formula, which reflects the fact that area is a two-dimensional measurement.

Composite (Compound) Shapes

A composite shape is made up of two or more basic shapes combined together, such as a rectangle attached to a triangle, or a rectangle with a semicircle on one end. To find the perimeter or area of a composite shape, break it down into its basic component shapes, calculate the perimeter or area of each part separately, and then add (or in some cases subtract, for shapes with a piece removed) the results together, taking care not to double-count any shared boundaries when finding the perimeter.

Practical Applications of Perimeter and Area

Perimeter and area calculations are essential in many real-life situations: calculating the amount of fencing needed to enclose a farm or garden (perimeter), determining how many tiles or how much carpet is needed to cover a floor (area), estimating the amount of paint needed to cover walls (area), calculating the cost of materials for construction projects based on measured dimensions, and determining the size of land plots for buying, selling, or farming purposes.

Common Mistakes in Perimeter and Area Calculations

Common errors in this topic include confusing perimeter and area formulas, or using the wrong units (forgetting that area should be expressed in square units, while perimeter uses simple length units), using a slanted side instead of the perpendicular height when calculating the area of a triangle or parallelogram, forgetting to double the radius to obtain the diameter (or vice versa) in circle formulas, and making errors when breaking down composite shapes into their basic component parts.

Summary

Perimeter is the total distance around the boundary of a shape, measured in units of length, while area is the amount of space enclosed within a shape, measured in square units. Different shapes have specific formulas: rectangles and squares use simple multiplication, triangles and parallelograms rely on base and perpendicular height, trapeziums use the average of the parallel sides multiplied by height, and circles use the constant π together with the radius. Composite shapes can be handled by breaking them into simpler shapes and combining the results, and these skills are widely applied in construction, farming, and everyday measurement tasks.

Strengthening Skills Through Practice

Perimeter and area problems become much easier with regular practice sketching and labelling shapes before calculating, since a clear diagram helps identify which measurements are needed and reduces the chance of applying the wrong formula. Practising a mix of straightforward and composite shape problems, along with word problems set in real contexts such as farming, flooring, and fencing, helps students build the confidence to select and apply the correct formula quickly and accurately.

Worked Examples

Example 1: Find the perimeter of a rectangle 18 cm by 9 cm. P = 2(18 + 9) = 2 × 27 = 54 cm.

Example 2: Find the area of a circle with radius 14 cm (use π = 22/7). A = (22/7) × 14² = (22/7) × 196 = 616 cm².

Example 3: Find the area of a triangle with base 16 cm and height 9 cm. A = ½ × 16 × 9 = 72 cm².

Example 4: Find the area of a trapezium with parallel sides 10 cm and 16 cm, and height 7 cm. A = ½ × (10 + 16) × 7 = ½ × 26 × 7 = 91 cm².

Example 5: A composite shape is made of a rectangle 12 cm by 8 cm with a right-angled triangle of base 8 cm and height 5 cm attached to one side. Find the total area. Rectangle area = 12 × 8 = 96 cm². Triangle area = ½ × 8 × 5 = 20 cm². Total area = 96 + 20 = 116 cm².

Student Exercise

Solve the following problems, showing all your working:

1. Find the perimeter of a rectangle 22 cm by 13 cm.
2. Find the circumference of a circle with radius 10.5 cm (use π = 22/7).
3. Find the area of a square with side 14 cm.
4. Find the area of a triangle with base 18 cm and height 10 cm.
5. Find the area of a parallelogram with base 15 cm and height 8 cm.
6. Find the area of a trapezium with parallel sides 9 cm and 15 cm, and height 6 cm.
7. Find the area of a circle with radius 21 cm (use π = 22/7).
8. A rectangular field measures 40 m by 25 m. Find the cost of fencing it at ₦500 per metre.
9. A composite shape is made of a rectangle 20 cm by 14 cm with a semicircle of diameter 14 cm attached to one end. Find its total area (use π = 22/7).
10. Find the perimeter of a triangle with sides 9 cm, 12 cm, and 15 cm.