Lesson Note
INTRODUCTION
Mensuration is the branch of mathematics that deals with the measurement of shapes and figures. It helps students understand how to calculate lengths, areas, and volumes of everyday objects such as rooms, gardens, and sports fields. By mastering mensuration, learners can solve practical problems like finding the amount of paint needed for a wall or the area of a football pitch. In this lesson, we will explore the basic formulas and techniques used to find perimeters, areas, and other measurements for common two‑dimensional shapes, as well as the properties of circles, arcs, sectors, and segments.
MAIN CONTENT
Rectangles and Squares
Rectangles are four‑sided figures with opposite sides equal and all angles right angles. A square is a special rectangle where all four sides are equal. The perimeter of a rectangle is the total length around it and is calculated by adding twice the length and twice the breadth. The area is the amount of surface it covers and is found by multiplying length by breadth. For a square, the perimeter is four times the side length and the area is the side length squared.
Example:
A rectangle has length $12$ cm and breadth $7$ cm.
Step 1: Perimeter $= 2(12+7)=2(19)=38$ cm.
Step 2: Area $= 12 \times 7=84$ cm$^{2}$.
Answer: Perimeter $38$ cm, Area $84$ cm$^{2}$.
Practice Question: Find the perimeter of a square with side $9$ cm.
Answer: $36$ cm.
Practice Question: Find the area of a rectangle $10$ cm by $6$ cm.
Answer: $60$ cm$^{2}$.
Triangles
A triangle is a figure with three sides. The area of any triangle can be calculated as one half the product of its base and the perpendicular height from that base. The perimeter is the sum of all three side lengths. Different types of triangles—equilateral, isosceles, and scalene—have different side relationships, but the area formula remains the same.
Example:
A triangle has base $8$ cm and height $5$ cm.
Step 1: Area $=\frac{1}{2}\times 8 \times 5=20$ cm$^{2}$.
Step 2: If the triangle’s sides are $7$, $9$, and $10$ cm, the perimeter $=7+9+10=26$ cm.
Answer: Area $20$ cm$^{2}$, Perimeter $26$ cm.
Practice Question: Find the area of a triangle with base $6$ cm and height $4$ cm.
Answer: $12$ cm$^{2}$.
Practice Question: Find the perimeter of a triangle with sides $5$, $12$, and $13$ cm.
Answer: $30$ cm.
Circles: Circumference and Area
A circle is defined by its radius $r$ (distance from centre to any point on the boundary) and diameter $d=2r$. The circumference is the distance around the circle, calculated as $C=2\pi r$ or $C=\pi d$. The area is the amount of surface inside the circle, given by $A=\pi r^{2}$.
Example:
A circle has radius $5$ cm.
Step 1: Circumference $=2\pi\times5=10\pi$ cm.
Step 2: Area $=\pi\times5^{2}=25\pi$ cm$^{2}$.
Answer: Circumference $10\pi$ cm, Area $25\pi$ cm$^{2}$.
Practice Question: Find the circumference of a circle with diameter $14$ cm.
Answer: $14\pi$ cm.
Practice Question: Find the area of a circle with radius $9$ cm.
Answer: $81\pi$ cm$^{2}$.
Arcs, Sectors and Segments of a Circle
An arc is a part of a circle’s circumference. The length of an arc depends on the central angle $\theta$ (in degrees) and the radius. The formula is $L=\frac{\theta}{360}\times 2\pi r$. A sector is a “slice” of a circle bounded by two radii and an arc; its area is $A=\frac{\theta}{360}\times \pi r^{2}$. A segment is the region between an arc and the chord that closes it; its area is the sector area minus the area of the isosceles triangle formed by the two radii and the chord. The triangle’s area is $\frac{1}{2}r^{2}\sin\theta$.
Example:
Circle radius $6$ cm, central angle $60^{\circ}$.
Step 1: Arc length $$L=\frac{60}{360}\times 2\pi\times6=\frac{1}{6}\times12\pi=2\pi$$ cm.
Step 2: Sector area $$A_{sector}=\frac{60}{360}\times \pi\times6^{2}=6\pi$$ cm$^{2}$.
Step 3: Triangle area $$A_{tri}=\frac{1}{2}\times6^{2}\times\sin60^{\circ}=18\times\frac{\sqrt{3}}{2}=9\sqrt{3}$$ cm$^{2}$.
Step 4: Segment area $A_{segment}=6\pi-9\sqrt{3}$ cm$^{2}$.
Answer: Arc length $2\pi$ cm, Sector area $6\pi$ cm$^{2}$, Segment area $6\pi-9\sqrt{3}$ cm$^{2}$.
Practice Question: Find the arc length of a circle radius $10$ cm with central angle $90^{\circ}$.
Answer: $5\pi$ cm.
Practice Question: Find the sector area of a circle radius $8$ cm with central angle $45^{\circ}$.
Answer: $2\pi$ cm$^{2}$.
Practice Question: Find the segment area of a circle radius $5$ cm with central angle $120^{\circ}$.
Answer: $5\pi- \frac{25\sqrt{3}}{2}$ cm$^{2}$.
Parallelograms and Trapeziums
A parallelogram has opposite sides equal and parallel. Its area is found by multiplying the base by the perpendicular height. A trapezium (or trapezoid) has two parallel sides; its area is one half the sum of the parallel sides multiplied by the height.
Example:
Parallelogram: base $10$ cm, height $4$ cm.
Step 1: Area $=10\times4=40$ cm$^{2}$.
Trapezium: bases $8$ cm and $12$ cm, height $5$ cm.
Step 2: Area $=\frac{1}{2}(8+12)\times5=20\times5=100$ cm$^{2}$.
Answer: Parallelogram area $40$ cm$^{2}$, Trapezium area $100$ cm$^{2}$.
Practice Question: Find the area of a parallelogram with base $7$ cm and height $3$ cm.
Answer: $21$ cm$^{2}$.
Practice Question: Find the area of a trapezium with bases $6$ cm and $10$ cm and height $4$ cm.
Answer: $32$ cm$^{2}$.
SUMMARY
Mensuration equips students with the tools to measure and calculate the properties of everyday shapes. From the simple rectangle and square to the more complex circle and its parts, each formula links a shape’s dimensions to its perimeter or area. By practicing the step‑by‑step calculations shown above, learners can confidently solve real‑world problems involving space and material estimation. Remember that the key to accuracy is correctly identifying the required measurement (perimeter, area, arc length, etc.) and applying the appropriate formula with the correct units.
EVALUATION
1. Compute the perimeter of a rectangle with length $15$ cm and breadth $9$ cm.
2. Find the area of a triangle with base $14$ cm and height $7$ cm.
3. Determine the circumference of a circle with radius $11$ cm.
4. Calculate the arc length of a circle radius $8$ cm with central angle $120^{\circ}$.
5. Find the area of a sector of a circle radius $6$ cm with central angle $90^{\circ}$.