4.9 Mensuration of 2D shapes

Learning objectives

4.9A Measurements within the metric systemconvert metric measurements, including linear and area units such as cm² and m² 4.9B Perimeters of triangles and rectanglesfind the perimeter of shapes made from triangles and rectangles 4.9C Areas of triangles and rectanglesfind the area of simple shapes using the formulae for the areas of triangles and rectangles 4.9D The area of parallelograms and trapeziafind the area of parallelograms and trapezia 4.9E Circumferences and areas of circles using relevantfind circumferences and areas of circles using relevant formulae; find perimeters and areas of semicircles 4.9HA Perimeters and areas of sectors of circles Radianfind perimeters and areas of sectors of circles Radian measure is excluded

A metric conversion factor acts once on a length but is squared for an area. Since $1\text{ m}=100\text{ cm}$ , it follows that $1\text{ m}^2=100^2\text{ cm}^2=10{,}000\text{ cm}^2$ .

Conversion	Length	Area
m to cm	multiply by $100$	multiply by $10{,}000$
cm to m	divide by $100$	divide by $10{,}000$
km to m	multiply by $1000$	multiply by $1{,}000{,}000$

Write the unit relationship first, raise its numerical factor to the power shown by the unit, then multiply toward smaller units or divide toward larger units. Include the converted unit in the answer.

Changing $1\text{ m}^2$ to $100\text{ cm}^2$ converts only one dimension; an area has two dimensions, so the factor must be squared.

Perimeter is the total length of the exposed outer boundary. Trace the outline once and add every outside edge; internal joins do not contribute.

Step	Action
1	mark the starting corner and trace clockwise
2	infer missing horizontal or vertical lengths from aligned totals
3	add only exposed edges in consistent units
4	check that the trace returns to the start

For shapes made from identical rectangles or triangles, shared edges disappear from the perimeter. A cost per metre is applied only after the perimeter has been found.

Do not add every side of every component: doing so double-counts internal shared boundaries.

Use $A=lw$ for a rectangle and $A=\frac12bh$ for a triangle, where $h$ is perpendicular to the chosen base $b$ .

Shape structure	Area strategy
joined pieces	split, calculate, then add
cut-out region	calculate the whole, then subtract
repeated tiles	area of one tile $\times$ number of tiles

Draw decomposition lines, label each base and perpendicular height, calculate in one unit, and preserve full precision before any coverage or cost decision. Round a required number of whole tins or tiles upward.

A sloping side is not a triangle's height unless it is perpendicular to the chosen base. Trigonometric area of a general triangle belongs to Topic 4.8, not this objective.

A parallelogram has area $A=bh$ . A trapezium with parallel sides $a$ and $b$ has area $A=\frac12(a+b)h$ , where $h$ is the perpendicular distance between them.

Symbol	Geometric meaning
$a,b$	the two parallel side lengths
$h$	perpendicular separation, not a sloping side
$\frac12(a+b)$	average width of the trapezium

Mark the parallel sides, identify or derive their perpendicular separation, substitute with consistent units, and rearrange the same formula if a missing length is required.

Do not average an arbitrary pair of sides, and do not use the sloping edge as $h$ unless a right-angle condition makes it perpendicular.

For radius $r$ and diameter $d=2r$ , circumference is $C=2\pi r=\pi d$ and area is $A=\pi r^2$ . A semicircle has half the circular arc and half the circular area.

Measure	Semicircle result
curved arc	$\pi r$
perimeter	$\pi r+2r$
area	$\frac12\pi r^2$

For a composite region, find each radius from the given diameters, then add or subtract the relevant circular and polygonal parts. Use exact $\pi$ during working and round only at the end.

The perimeter of a semicircle includes its straight diameter; the area does not. Do not confuse $\pi r^2$ with $2\pi r$ .

A sector with central angle $\theta$ degrees is the fraction $\theta/360$ of a circle. Arc length is $L=\frac{\theta}{360}(2\pi r)$ and area is $A=\frac{\theta}{360}(\pi r^2)$ .

Required boundary	Include
arc length only	curved arc $L$
sector perimeter	curved arc $L$ plus two radii $2r$
major sector	use $360^\circ-\theta$ when the given angle describes the minor sector

Identify the centre, radius and intended minor or major angle, calculate the circle fraction in degrees, then include only the requested boundary or area. Radian measure is outside this syllabus objective.

A segment is bounded by an arc and a chord, while a sector is bounded by an arc and two radii; their perimeters and areas are not interchangeable.

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4.9 Mensuration of 2D shapes

Learning objectives

Convert metric lengths and areas

Find perimeters of compound rectilinear shapes

Find areas of triangles and rectangles

Find areas of parallelograms and trapezia

Find circle and semicircle measures

Find perimeters and areas of sectors

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