Year 10 Measurement and surds: revision mind map

1Irrational numbers and surds

A surd is a root that isn't a whole number, like $\sqrt2$. Simplify using square factors.

$\sqrt8 = 2\sqrt2$   $\sqrt{48} = \sqrt{16}\sqrt3 = 4\sqrt3$

2Operations with surds

Add and subtract like surds only. Multiply or divide the numbers outside and inside separately.

$\sqrt{12} + \sqrt{27} = 2\sqrt3 + 3\sqrt3 = 5\sqrt3$

$\sqrt6 \times \sqrt3 = \sqrt{18} = 3\sqrt2$   $6\sqrt{10} \div 2\sqrt5 = 3\sqrt2$

3Rationalising the denominator

Multiply top and bottom so there's no surd on the bottom.

$\dfrac{3}{\sqrt2} = \dfrac{3\sqrt2}{2}$

$\dfrac{1}{\sqrt3 + 1} \times \dfrac{\sqrt3 - 1}{\sqrt3 - 1} = \dfrac{\sqrt3 - 1}{2}$

4Review of length

Circumference $2\pi r$. Arc length $\tfrac{\theta}{360} \times 2\pi r$.

$r = 10,\ \theta = 72^\circ$: arc $\tfrac15 \times 20\pi = 4\pi \approx 12.57$

5Pythagoras in 3D

Find a right triangle, or use $d = \sqrt{a^2 + b^2 + c^2}$ for a box.

Box 2 by 3 by 6: $\sqrt{4 + 9 + 36} = 7$

6Review of area

Triangle $\tfrac12 bh$, trapezium $\tfrac12(a + b)h$, circle $\pi r^2$, sector $\tfrac{\theta}{360}\pi r^2$

Sector $r = 6,\ \theta = 60^\circ$: $\tfrac16 \times 36\pi = 6\pi \approx 18.85$

7Measurement errors and accuracy

The error is up to half the smallest unit. Percentage error = error ÷ measurement × 100%.

4.2 m to the nearest 0.1 m: between 4.15 m and 4.25 m

Percentage error $\tfrac{0.05}{4.2} \times 100\% \approx 1.2\%$

8Surface area: prisms and cylinders

Add every face. Cylinder: $2\pi r^2 + 2\pi rh$.

Cylinder $r = 4,\ h = 10$: $32\pi + 80\pi = 112\pi \approx 351.9$

9Surface area: pyramids and cones

Cone: $\pi r^2 + \pi r s$, where $s$ is the slant height.

$r = 3,\ h = 4$: $s = 5$, so $SA = 9\pi + 15\pi = 24\pi \approx 75.4$

Watch out: use the slant height, not the vertical height.

10Volume: prisms and cylinders

$V = Ah$. Cylinder: $V = \pi r^2 h$.

$r = 5,\ h = 8$: $V = 200\pi \approx 628.3$

11Volume: pyramids and cones

$V = \tfrac13 Ah$: a third of the matching prism.

Cone $r = 3,\ h = 4$: $\tfrac13 \times 9\pi \times 4 = 12\pi \approx 37.7$

Square pyramid, base 6, height 10: $\tfrac13 \times 36 \times 10 = 120$

12Spheres

$SA = 4\pi r^2$ and $V = \tfrac43 \pi r^3$

$r = 6$: $SA = 144\pi \approx 452.4$

$V = \tfrac43\pi(216) = 288\pi \approx 904.8$