Hello, and welcome to today's mathematics lesson. We are going to explore the area of a trapezium and various polygons. By the end of this lesson, you will understand how to find areas and perimeters of various plane figures — triangles, rectangles, squares, trapeziums, parallelograms, rhombuses, and circles. You will also learn how to apply these formulas to solve practical problems.
Let us begin with some essential definitions. The perimeter of any plane figure is simply the total length of its boundary. The area, on the other hand, is the amount of surface enclosed within that boundary.
Before we dive into formulas, let us quickly review the units we use. For perimeter, we use metres, centimetres, or decimetres. Remember, one metre equals one hundred centimetres. For area, we use square units — square metres, written as m², square centimetres as cm², square millimetres as mm², and so on. One square metre equals ten thousand square centimetres.
Now, let us revisit triangles. There are two powerful ways to find the area of a triangle.
First, if you know all three sides — let us call them a, b, and c — you can use Heron's formula. Calculate the semi-perimeter, which is half of the perimeter: s = (a + b + c)/2. Then the area equals the square root of s(s-a)(s-b)(s-c). This formula is known as Heron's formula.
Second, if you know one side and the perpendicular height to that side, the area is simply half times base times height: ½ × b × h. Any side can be your base — just make sure you use the correct corresponding height.
Here is a worked example. Suppose a triangle has sides 9 centimetres, 12 centimetres, and 15 centimetres. The semi-perimeter is 18 centimetres. Using Heron's formula, the area becomes the square root of 18 × 9 × 6 × 3, which equals 54 cm². To find the altitude to the longest side, use the area formula backwards: height equals twice the area divided by base, giving 7.2 centimetres.
Moving on to rectangles. The perimeter of a rectangle is twice the sum of its length and breadth: 2(l + b). The area is simply length times breadth. If you need the diagonal, apply Pythagoras theorem: d² = l² + b².
For squares, everything simplifies beautifully. Perimeter equals four times the side. Area equals side squared. The diagonal equals side times √2, which is approximately 1.414 times the side.
Let us see this in action. A square has perimeter 28 centimetres. Each side must be 7 centimetres. The area is 49 cm², and the diagonal is 7 times √2, which is approximately 9.898 centimetres.
Now we reach the heart of this chapter — the trapezium. A trapezium is a quadrilateral with one pair of parallel sides.
The area of a trapezium equals half times the sum of the parallel sides times the height: ½(a + b) × h. Here, a and b are the two parallel sides, and h is the perpendicular distance between them.
Let us work through an example. The parallel sides of a trapezium are in the ratio 3 to 5. The distance between them is 10 centimetres, and the area is 120 square centimetres. Calling the sides 3x and 5x, we set up the equation: 120 equals half times 8x times 10. Solving gives x = 3, so the parallel sides are 9 centimetres and 15 centimetres.
Another useful problem: if you know the perimeter and the non-parallel sides, you can find the sum of parallel sides by subtraction. With perimeter 52 centimetres and non-parallel sides 9 and 11 centimetres, the sum of parallel sides is 32 centimetres. At height 8 centimetres, the area becomes half times 32 times 8, which is 128 square centimetres.
Next, the parallelogram. The area equals base times height — but remember, you must use the perpendicular height, not the slant side. A key insight: either pair of parallel sides can serve as base, with their corresponding heights.
Here is a clever application. A parallelogram has sides 12 centimetres and 8 centimetres. The distance between the 12 centimetre sides is 5 centimetres. The area is 60 square centimetres. Now, using the 8 centimetre side as base, the corresponding height must be 60 divided by 8, which is 7.5 centimetres.
Also remember, each diagonal divides a parallelogram into two equal triangles. This means you can sometimes use triangle area formulas to find parallelogram areas.
The rhombus is a special parallelogram with all sides equal. Its perimeter is four times the side length.
Most importantly, the area equals half the product of its diagonals: ½ × d₁ × d₂. This works because the diagonals bisect each other at right angles. Each diagonal also bisects the rhombus into two equal triangles, and all four triangles formed by the diagonals are congruent.
Consider a rhombus with diagonals 16 centimetres and 12 centimetres. The area is half times 16 times 12, giving 96 square centimetres. Since the diagonals bisect at 90 degrees, each half-diagonal forms a right triangle with side as hypotenuse: 8 centimetres and 6 centimetres give side length 10 centimetres by Pythagoras theorem. The perimeter is 40 centimetres. Using area equals base times height, the height is 96 divided by 10, which is 9.6 centimetres.
Finally, we come to the circle. A circle is the set of all points at a fixed distance — the radius — from a centre point.
The diameter is twice the radius. The diameter is twice the radius, so d = 2r. The diameter is twice the radius, so d = 2r. The circumference equals 2πr or πd. The area equals πr². Unless specified otherwise, use π = 22/7.
Let us connect this with other shapes. A square with side 22 centimetres has perimeter 88 centimetres. If a circle has the same circumference, then 2πr equals 88, giving radius r = 14 centimetres. The circle's area is πr², that is 22/7 × 14², which equals 616 cm².
For circular paths between concentric circles, subtract the inner area from the outer area. If inner circumference is 176 metres, the inner radius is 28 metres. With path width 3.5 metres, outer radius is 31.5 metres. The path area is π × 31.5² − π × 28², which equals 3118.5 − 2464 square metres, giving 654.5 m².
Let us recap the key takeaways from this lesson.
First, Heron's formula lets you find a triangle's area when all three sides are known: area equals the square root of s(s-a)(s-b)(s-c), where semi-perimeter s = (a+b+c)/2.
Second, the area of a trapezium is ½(a + b) × h: half the sum of parallel sides times the height.
Third, a parallelogram's area is base times height, using the perpendicular distance between parallel sides.
Fourth, a rhombus area is ½ × d₁ × d₂: half the product of its diagonals, which bisect each other at right angles.
Fifth, circle circumference is 2πr or πd, and area is πr², where π = 22/7 unless specified otherwise.
Sixth, for circular paths between concentric circles, subtract inner area from outer area; for other composite shapes, break into simpler figures and combine areas as needed.
You have now learned powerful tools for measuring plane figures. Practice applying these formulas flexibly — sometimes working forwards from given dimensions, sometimes backwards from known areas. Keep exploring, keep calculating, and enjoy discovering how geometry describes the world around you. Until next time, happy learning!