Perimeter and Area of Basic Shapes

Perimeter and area are two fundamental measurements used to describe plane figures — flat, two-dimensional shapes. The perimeter of a shape is the total length of its boundary, calculated by adding up the lengths of all its sides. Area, on the other hand, measures the amount of surface enclosed within that boundary. Both concepts are essential in everyday life: builders calculate areas to order flooring materials, engineers compute perimeters when fencing land, and architects rely on these measurements constantly when designing spaces. In this topic, we focus on four basic plane figures: rectangles, triangles, parallelograms, and trapeziums (also called trapezoids in some countries). Each shape has its own dedicated formula for both perimeter and area. Understanding where these formulas come from — not just memorising them — is key to applying them correctly in unfamiliar problems and to building a solid foundation for more advanced geometry. All perimeter measurements are expressed in linear units (e.g., cm, m, km), while area measurements are expressed in square units (e.g., cm², m², km²). This distinction matters: if you double every side of a rectangle, the perimeter doubles but the area quadruples. Keeping track of units is not just a formality — it reveals the underlying mathematics of how shapes scale.

Key takeaways
  • Rectangle
  • Triangle
  • Parallelogram
  • Trapezium

Rectangle

A rectangle has four right angles with opposite sides equal. If the length is l and the width is w, then the perimeter is P = 2(l + w) because you travel along each pair of parallel sides twice. The area is A = l × w, which counts how many unit squares fit inside. A square is a special rectangle where l = w, giving P = 4l and A = l².

Triangle

A triangle has three sides, so its perimeter is simply P = a + b + c, where a, b, and c are the side lengths. The area formula is A = ½ × base × height, written A = ½bh. The height (h) must be the perpendicular distance from the base to the opposite vertex — NOT the slant side. For a right-angled triangle the two shorter sides serve as base and height directly. This formula is derived from the fact that any triangle is exactly half of a parallelogram with the same base and height.

Parallelogram

A parallelogram has two pairs of parallel sides. If the parallel sides have lengths a and b, the perimeter is P = 2(a + b). The area is A = base × height = bh, where h is the perpendicular height between the two parallel sides of length b, not the slant side. This formula is justified by slicing and rearranging: if you cut a right triangle off one end of the parallelogram and reattach it to the other end, you form a rectangle with dimensions b × h.

Trapezium

A trapezium (trapezoid) has exactly one pair of parallel sides, called the parallel sides or bases, with lengths a and b. The other two sides (the legs) have lengths c and d. The perimeter is P = a + b + c + d. The area formula is A = ½(a + b)h, where h is the perpendicular distance between the two parallel sides. This formula averages the two parallel sides and multiplies by the height — intuitively, it is the area of a rectangle whose width is the average of the two bases. A parallelogram is a special trapezium where a = b, which correctly gives A = ah = bh.

The Key Equation

Atrapezium=12(a+b)hA_{\text{trapezium}} = \frac{1}{2}(a + b)h
  • A — Area (cm² (or m², etc.)): The surface enclosed inside the trapezium.
  • a — First parallel side (base 1) (cm): The length of one of the two parallel sides.
  • b — Second parallel side (base 2) (cm): The length of the other parallel side.
  • h — Perpendicular height (cm): The perpendicular distance between the two parallel sides — not the slant leg.

Worked Example

Problem. A trapezium has parallel sides of length 8 cm and 14 cm. Its perpendicular height is 5 cm, and its two non-parallel (slant) sides are each 6.5 cm long. Calculate (a) the perimeter and (b) the area of the trapezium.

P=35 cm,A=55 cm2P = 35\text{ cm}, \quad A = 55\text{ cm}^2

Answer. Perimeter = 35 cm; Area = 55 cm²

Figure
Diagram
Perimeter and Area of Basic Plane FiguresRectanglel = 10 cm, w = 6 cmP = 2(10+6) = 32 cmA = 10×6 = 60 cm²10 cm6 cmTriangleb = 100 px, h = 60 pxA = ½ × b × hhbParallelogramA = base × heighthbaseTrapeziuma (top), b (bottom)A = ½(a+b)×hhabSummary of FormulasRectangle: P=2(l+w), A=lwTriangle: P=a+b+c, A=½bhParallelogram: P=2(a+b), A=bhTrapezium: P=a+b+c+d, A=½(a+b)h
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-325–-265 · Ancient Greek

Euclid systematised plane geometry in his landmark work 'Elements' (c. 300 BCE), a 13-volume treatise that rigorously proved results about areas and perimeters of polygons using axioms and logical deduction. His proof that the area of a parallelogram equals base times height, and that a triangle is half of a parallelogram with the same base and height (Book I, Propositions 35–37), provided the logical foundation for the area formulas still taught today. 'Elements' was used as the standard geometry textbook for over two thousand years. When King Ptolemy I asked Euclid if there was a shorter path to learning geometry than studying 'Elements', Euclid reportedly replied: 'There is no royal road to geometry.' This phrase is still used to mean that mastery requires effort, regardless of your status.

  • Circumference and Area of a Circle — Extends perimeter and area ideas to curved shapes, introducing π as the key constant linking a circle's diameter to its circumference and area.
  • Surface Area and Volume of 3D Shapes — Surface area calculations for prisms and pyramids rely directly on computing the areas of the plane figures covered in this topic.
  • Pythagoras' Theorem — Often needed alongside area formulas to find missing side lengths or perpendicular heights in triangles and other polygons before computing area.
  • Area of Composite Figures — Real-world shapes are often combinations of rectangles, triangles, and trapeziums; this topic applies the basic area formulas together to find areas of complex shapes.