Mensuration

The One Thing

Mensuration concerns the measurement of boundaries, enclosed regions, and the outer surfaces of solids. Reliable solutions depend on identifying the figure and required measurement, using consistent units, selecting the appropriate formula, and stating the answer with the correct unit.

Definitions and Results

• Perimeter: The total length of the boundary of a closed plane figure.
• Area: The region enclosed by a plane figure, measured in square units.
• Surface Area: The total area of all outer surfaces of a three-dimensional solid.
• Lateral or Curved Surface Area: The area of the side surface of a solid, excluding its base or bases.
• Total Surface Area: The sum of the areas of all surfaces, including the base or bases.
• Circumference: The perimeter of a circle.
• Radius: The distance from the centre of a circle to any point on its circumference.
• Diameter: A line segment passing through the centre of a circle and joining two points on its circumference; it is twice the radius.
• Slant Height: The inclined height of a cone measured along its curved surface from the vertex to the edge of the base.
• Net: A two-dimensional pattern that can be folded to form a three-dimensional solid.
• Unit Conversion: Changing a measurement into an equivalent unit, such as centimetres to metres or square centimetres to square metres.

Plane Figures

• Rectangle:

Perimeter $=2(\text{length}+\text{breadth})$ Area $=\text{length}\times \text{breadth}$

• Square:

Perimeter $=4\times \text{side}$ Area $={\text{side}}^2$

• Triangle:

Area $=\frac{1}{2}\times \text{base}\times \text{perpendicular height}$

• Parallelogram:

Area $=\text{base}\times \text{perpendicular height}$

• Rhombus:

Area $=\frac{1}{2}\times \text{product of its diagonals}$

• Trapezium:

Area $=\frac{1}{2}\times \text{sum of parallel sides}\times \text{perpendicular distance between them}$

• Circle:

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

• Semicircle:

Area $=\frac{1}{2}\pi r^2$ Perimeter $=\pi r+2r$, because it includes the curved part and the diameter.

Three-Dimensional Solids

• Cuboid with length $l$, breadth $b$, and height $h$:

Lateral surface area $=2h(l+b)$ Total surface area $=2(lb+bh+hl)$ Volume $=lbh$

• Cube with side $a$:

Lateral surface area $=4a^2$ Total surface area $=6a^2$ Volume $=a^3$

• Cylinder:

Curved surface area $=2\pi rh$ Total surface area $=2\pi r(h+r)$ Volume $=\pi r^{2}h$

• Cone:

Curved surface area $=\pi rl$, where $l$ is the slant height Total surface area $=\pi r(l+r)$ Volume $=\frac{1}{3}\pi r^{2}h$ If radius, perpendicular height, and slant height form a right triangle, then $$l^2=r^2+h^2.$$

• Sphere:

Surface area $=4\pi r^2$ Volume $=\frac{4}{3}\pi r^3$

• Hemisphere:

Surface area excluding its base $=2\pi r^2$ Total surface area $=3\pi r^2$ Volume $=\frac{2}{3}\pi r^3$

Units and Conditions

• Perimeter and circumference use linear units such as $\text{cm}$ or $\text{m}$.
• Area and surface area use square units such as ${\text{cm}}^2$ or ${\text{m}}^2$.
• Volume uses cubic units such as ${\text{cm}}^3$ or ${\text{m}}^3$.
• Before applying a formula, convert all dimensions into the same unit.
• Use $\pi =\frac{22}{7}$ or $3.14$ when a value is not specified; choose the value that simplifies the calculation when appropriate.

Worked Methods

1. Measuring a Rectangle or Square

• Identify whether the required measurement is perimeter or area.
• For a rectangle, use:

$$P=2(l+b),A=lb.$$

• For a square, use:

$$P=4a,A=a^2.$$

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

2. Finding the Area of a Triangle or Parallelogram

• Identify the base.
• Use the perpendicular height, not a slant or inclined side.
• For a triangle, calculate:

$$A=\frac{1}{2}bh.$$

• For a parallelogram, calculate:

$$A=bh.$$

• Express the result in square units.

3. Finding the Area of a Rhombus or Trapezium

• Identify the relevant diagonals of the rhombus, or the parallel sides and perpendicular distance of the trapezium.
• For a rhombus, use:

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

• For a trapezium, use:

$$A=\frac{1}{2}(a+b)h,$$ where $a$ and $b$ are the parallel sides and $h$ is their perpendicular distance.

• Substitute the measurements and write the answer in square units.

4. Finding the Circumference or Area of a Circle

• Identify the radius $r$ or diameter $d$.
• If the diameter is given, use $d=2r$ when the formula requires the radius.
• For circumference, use:

$$C=2\pi r=\pi d.$$

• For area, use:

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

• Use $\pi =\frac{22}{7}$ or $3.14$ when appropriate.
• Give circumference in linear units and area in square units.

5. Finding the Area or Perimeter of a Semicircle

• Identify the radius $r$.
• For area, use:

$$A=\frac{1}{2}\pi r^2.$$

• For perimeter, include both the curved part and the diameter:

$$P=\pi r+2r.$$

• Do not use the curved length alone when the question asks for the perimeter.

6. Finding the Surface Area or Volume of a Cuboid or Cube

• Identify whether the solid is a cuboid or a cube.
• For a cuboid with dimensions $l$, $b$, and $h$, select:

$$\text{LSA}=2h(l+b),$$ $$\text{TSA}=2(lb+bh+hl),$$ $$V=lbh.$$

• For a cube with side $a$, select:

$$\text{LSA}=4a^2,$$ $$\text{TSA}=6a^2,$$ $$V=a^3.$$

• Use lateral surface area when the bases are excluded and total surface area when all surfaces are included.
• State surface area in square units and volume in cubic units.

7. Finding the Surface Area or Volume of a Cylinder

• Identify the radius $r$ and height $h$.
• For the curved surface area, use:

$$\text{CSA}=2\pi rh.$$

• For the total surface area, include both circular bases:

$$\text{TSA}=2\pi r(h+r).$$

• For volume, use:

$$V=\pi r^{2}h.$$

• State the appropriate square or cubic unit.

8. Finding the Surface Area or Volume of a Cone

• Identify the radius $r$, perpendicular height $h$, and slant height $l$.
• If $l$ is not given but $r$ and $h$ are known, calculate it using:

$$l^2=r^2+h^2.$$

• For curved surface area, use:

$$\text{CSA}=\pi rl.$$

• For total surface area, use:

$$\text{TSA}=\pi r(l+r).$$

• For volume, use the perpendicular height $h$:

$$V=\frac{1}{3}\pi r^{2}h.$$

• Distinguish carefully between $l$, the slant height, and $h$, the perpendicular height.

9. Finding the Surface Area or Volume of a Sphere or Hemisphere

• Identify the radius $r$.
• For a sphere, use:

$$\text{Surface area}=4\pi r^2,V=\frac{4}{3}\pi r^3.$$

• For a hemisphere excluding its base, use:

$$\text{Surface area}=2\pi r^2.$$

• For the total surface area of a hemisphere, include the circular base:

$$\text{TSA}=3\pi r^2.$$

• For the volume of a hemisphere, use:

$$V=\frac{2}{3}\pi r^3.$$

10. Solving Problems Involving Nets

• Identify every visible region in the net.
• Determine which regions form the surfaces of the solid.
• Calculate the area of each region.
• Add the areas of all visible regions:

$$\text{Total surface area}=\text{sum of the areas of all surfaces}.$$

• State the result in square units.

11. Solving Composite-Figure and Path Problems

• Divide a composite figure into familiar figures.
• Calculate the individual areas or perimeters.
• Combine the results carefully, avoiding double-counting shared boundaries.
• For a path or border, calculate:

$$\text{Area of path}=\text{area of outer region}-\text{area of inner region}.$$

• For practical problems involving painting, polishing, covering, fencing, flooring, or wrapping, identify whether the required measurement is perimeter, area, or surface area.

Where It Goes Wrong

• Using a slant or inclined side instead of the perpendicular height in the area formulas for triangles and parallelograms.
• Confusing lateral or curved surface area, which excludes the base or bases, with total surface area, which includes them.
• Omitting the diameter when finding the perimeter of a semicircle; its perimeter is $\pi r+2r$, not merely $\pi r$.
• Using the cone’s slant height $l$ in the volume formula instead of the perpendicular height $h$; the volume is $\frac{1}{3}\pi r^{2}h$.
• Failing to convert all dimensions into the same unit before applying a formula.
• Giving a numerical result without the required linear, square, or cubic unit, or failing to check whether its size and dimensions are reasonable.

What Gets Asked

This material supports questions requiring students to:

• Calculate the perimeter or area of rectangles, squares, triangles, parallelograms, rhombuses, trapeziums, circles, and semicircles.
• Find the circumference or area of a circle using a given radius or diameter.
• Calculate the lateral, curved, or total surface area and volume of cuboids, cubes, cylinders, cones, spheres, and hemispheres.
• Determine a cone’s slant height using $l^2=r^2+h^2$.
• Interpret a net and add the areas of its visible regions to obtain total surface area.
• Solve composite-figure problems by dividing shapes into familiar figures.
• Find the area of a path or border by subtracting the inner area from the outer area.
• Apply mensuration to practical contexts involving painting, polishing, covering, fencing, flooring, or wrapping.
• Convert units before calculation and select an appropriate value of $\pi$, using $\frac{22}{7}$ or $3.14$ when necessary.
• Distinguish between perimeter, area, surface area, and volume and attach the correct units to the final answer.