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SPI Mathematics · Formula sheet

Mensuration 2D formulas

9 formulas and 20 common traps for SPI Mathematics Mensuration 2D, grouped by subtopic.

Full notes with worked examples

Rectangles, Squares and Triangles

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Perimeter, area and cost of a rectangle

Rectangle and square

P=2(l+b),A=lb,Psquare=4a,Asquare=a2P = 2(l + b), \quad A = lb, \quad P_{\text{square}} = 4a, \quad A_{\text{square}} = a^2
  • l,bl, blength and breadth
  • aaside of the square

Diagonals of rectangles and squares

Diagonals

d=l2+b2,dsquare=a2,Asquare=d22d = \sqrt{l^2 + b^2}, \qquad d_{\text{square}} = a\sqrt{2}, \qquad A_{\text{square}} = \frac{d^2}{2}

Triangles and the regular hexagon

Triangle and hexagon

A=12bh,Aequilateral=34a2,Ahexagon=6×34a2=332a2A = \frac{1}{2}bh, \quad A_{\text{equilateral}} = \frac{\sqrt{3}}{4}a^2, \quad A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2

Common traps

Perimeter where area is needed

Fencing and walking round use the perimeter; carpeting, tiling and painting use the area. Decide which before you multiply by the rate.

Forgetting to halve the perimeter

A perimeter of 30 cm means l+b=15l + b = 15 cm, not 30 cm. Divide by 2 before splitting the sum in a ratio.

Counting rectangles with a fixed perimeter

Infinitely many rectangles have a perimeter of 14 cm. Only if the sides must be whole centimetres is the answer a count: l+b=7l + b = 7 gives 1 by 6, 2 by 5 and 3 by 4, so three.

Mixing metres and centimetres

A hall 0.5 m by 30 cm must become 50 cm by 30 cm before squaring. Options often print the same number in both units, so check the unit too.

Doubling the diagonal doubles the area

It does not. A square's area is d22\frac{d^2}{2}, so twice the diagonal gives four times the area.

Side or perimeter from an area

A square of area 81 square cm has side 81=9\sqrt{81} = 9 cm. Its perimeter, 36 cm, is the usual wrong option.

Forgetting the half

A triangle's area is half of base times height. Base 10 cm and height 6 cm give 30 square cm, not 60.

Too few triangles in a hexagon

A regular hexagon is six equilateral triangles, so multiply one triangle's area by 6. A hexagon of side 3 cm is 6×934=27326 \times \frac{9\sqrt{3}}{4} = \frac{27\sqrt{3}}{2} square cm, not 934\frac{9\sqrt{3}}{4}.

Circumference and Area of a Circle

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Circumference of a circle

Circumference

C=2πr=πdC = 2\pi r = \pi d

Wheels: distance and revolutions

Wheel

distance=n×2πr,n=distance2πr\text{distance} = n \times 2\pi r, \qquad n = \frac{\text{distance}}{2\pi r}
  • nnnumber of revolutions

Area of a circle and a semicircle

Area of a circle

A=πr2,Asemicircle=12πr2A = \pi r^2, \qquad A_{\text{semicircle}} = \frac{1}{2}\pi r^2

A wire bent from one shape into another

Same wire, new shape

2πr=4a,AcircleAsquare=4π=14112\pi r = 4a, \qquad \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{4}{\pi} = \frac{14}{11}

Common traps

The perimeter of a semicircle

A semicircle's boundary is half the circumference plus the straight diameter: πr+2r\pi r + 2r. Leaving out the diameter gives only the curved arc.

Circumferences add by radius, areas do not

If two circumferences are added, the radii simply add: 3 cm and 4 cm give 7 cm. If two areas are added, the new radius is r12+r22\sqrt{r_1^2 + r_2^2}: 3 cm and 4 cm give 5 cm, not 7 cm.

Kilometres against centimetres

1 km is 100 000 cm, not 1000 cm. Turn the distance into the unit of the circumference before you divide.

Radius or diameter of the wheel

Distance per turn gives πd\pi d. Dividing by π\pi gives the diameter; halve it only if the radius is asked.

Squaring the diameter

A circle of diameter 14 cm has area π×72=154\pi \times 7^2 = 154, not π×142=616\pi \times 14^2 = 616. Using the diameter makes the area four times too big.

A semicircle given the full circle's area

A semicircular top is half of πr2\pi r^2. The full circle's area is usually a printed wrong option.

Keeping the area instead of the length

When a wire is bent, its length stays the same, not the area it encloses. Equate the perimeters, never the areas.

Turning the area ratio round

Circle to square is 14 : 11 because the circle encloses more. If the question asks square to circle, the answer is 11 : 14.

Sectors and Segments

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Area of a sector, the clock hand and segments

Sector and segment

Asector=θ360∘πr2,Asegment=Asector−AtriangleA_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2, \qquad A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}
  • θθangle of the sector at the centre, in degrees

Arc length and the perimeter of a sector

Arc and perimeter

l=θ360∘×2πr,P=2r+l,A=12lrl = \frac{\theta}{360^\circ} \times 2\pi r, \qquad P = 2r + l, \qquad A = \frac{1}{2}lr
  • lllength of the arc

Common traps

Minute hand or hour hand

The minute hand turns 6∘6^\circ a minute; the hour hand only 0.5∘0.5^\circ a minute. In 20 minutes the minute hand sweeps 120∘120^\circ, a third of the circle.

Giving the sector when the segment is asked

A segment is the sector minus the triangle formed by the two radii and the chord. A quarter circle of radius 21 cm is 346.5 square cm, but its segment is only 346.5−220.5=126346.5 - 220.5 = 126 square cm.

Forgetting the second radius

A sector's perimeter has two straight edges. Arc == perimeter −2r- 2r, not perimeter −r- r.

Hunting for the angle

If the arc is known, the area is 12lr\frac{1}{2}lr at once. There is no need to find the angle first.

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