Geometry: angles, triangles and circles

Answer: $120$

Co-interior angles at parallel lines add to $180^\circ\mathclose{:}$ $180 - 60 = 120\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took co-interior angles to be equal → $60$

Answer: $24$

The two legs are the base and the height. The other leg is $\sqrt{10^2 - 6^2} = 8\mathclose{,}$ so the area is $\dfrac{1}{2} \times 6 \times 8 = 24\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to halve → $48$
• Used the hypotenuse as the height → $30$

Answer: $\pi$

The arc is $\dfrac{30}{360} = \dfrac{1}{12}$ of the circumference $2\pi r\mathclose{:}$ $\dfrac{1}{12} \times 2\pi \times 6 = \pi\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the area of the sector instead → $3 \pi$
• Used the diameter in place of the radius → $2 \pi$

Answer: $47$

The other two angles are equal and share what is left of $180^\circ\mathclose{:}$ $\dfrac{180 - 86}{2} = 47\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not halve what was left → $94$
• Halved the top angle instead → $43$

Answer: $165$

Each exterior angle is $\dfrac{360}{24} = 15^\circ\mathclose{,}$ and an interior and an exterior angle make a straight line: $180 - 15 = 165\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the exterior angle instead → $15$
• Gave the sum of all the interior angles → $3960$

Answer: $82$

Corresponding angles at parallel lines are equal: $82^\circ\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the two angles to add up to $180^\circ$ → $98$

Answer: $5$

Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs. $3^2 + 4^2 = 25\mathclose{,}$ and $\sqrt{25} = 5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the legs instead of their squares → $7$
• Subtracted the squares, which finds a LEG, not the hypotenuse → $\sqrt{7}$

Answer: $9 \pi$

Area $= \pi r^2 = \pi \times 3^2 = 9\pi\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the circumference formula $2\pi r$ → $6 \pi$
• Used the diameter in place of the radius → $36 \pi$

Answer: $72$

There are $4 + 5 + 6 = 15$ equal parts in $180^\circ\mathclose{,}$ so one part is $12^\circ$ and the largest angle is $6 \times 12 = 72\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the size of one part → $12$
• Shared out $360^\circ$ instead of $180^\circ$ → $144$

Answer: $6$

The exterior angles of any polygon add to $360^\circ\mathclose{,}$ so there are $\dfrac{360}{60} = 6$ of them — one per side.

Common mistakes, and the answer each one gives:

• Divided $180^\circ$ by the angle instead of $360^\circ$ → $3$

Answer: $72$

Vertically opposite angles are equal, so it is also $72^\circ\mathclose{.}$ (The two angles NEXT to it are $108^\circ$ each.)

Common mistakes, and the answer each one gives:

• Gave an angle next to it instead → $108$

Answer: $4$

Rearrange Pythagoras: the missing leg squared is $5^2 - 3^2 = 16\mathclose{,}$ so the leg is $\sqrt{16} = 4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the lengths instead of their squares → $2$
• Added the squares, which is the rule for the HYPOTENUSE → $\sqrt{34}$

Answer: $10 \pi$

Circumference $= 2\pi r = 2 \times \pi \times 5 = 10\pi\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the area formula $\pi r^2$ → $25 \pi$
• Used the diameter in place of the radius → $20 \pi$

Answer: $289$

Angles round a point add to $360^\circ\mathclose{:}$ $360 - 29 - 42 = 289\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $180^\circ$ instead of $360^\circ$ → $109$
• Took only $29^\circ$ away from $360^\circ$ → $331$

Answer: $39$

The three angles of any triangle add to $180^\circ\mathclose{,}$ so the third is $180 - 61 - 80 = 39\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $360^\circ\mathclose{,}$ the angles round a POINT → $219$
• Subtracted only one of the two given angles → $119$

Answer: $3240$

A polygon with $20$ sides splits into $18$ triangles from one corner, and each triangle contributes $180^\circ\mathclose{:}$ $(20 - 2) \times 180 = 3240\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied the number of sides by 180 without subtracting 2 → $3600$
• Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

Answer: $143$

Angles on a straight line add to $180^\circ\mathclose{:}$ $180 - 37 = 143\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $360^\circ\mathclose{,}$ the angles round a point → $323$
• Used $90^\circ\mathclose{,}$ as if the angles made a right angle → $53$

Answer: $40$

The perimeter needs all three sides. The hypotenuse is $\sqrt{8^2 + 15^2} = \sqrt{289} = 17\mathclose{,}$ so the perimeter is $8 + 15 + 17 = 40\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added only the two legs → $23$
• Used $a^2 + b^2$ for the hypotenuse, without the root → $312$

Answer: $\frac{3 \pi}{4}$

The sector is $\dfrac{30}{360} = \dfrac{1}{12}$ of the circle, so its area is $\dfrac{1}{12} \times \pi \times 3^2 = \frac{3 \pi}{4}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the area of the whole circle → $9 \pi$
• Used $2\pi r$ (the arc's formula) instead of $\pi r^2$ → $\frac{\pi}{2}$

Answer: $127$

Co-interior angles at parallel lines add to $180^\circ\mathclose{:}$ $180 - 53 = 127\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took co-interior angles to be equal → $53$

Answer: $76$

The other two angles are equal and share what is left of $180^\circ\mathclose{:}$ $\dfrac{180 - 28}{2} = 76\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not halve what was left → $152$
• Halved the top angle instead → $14$

Answer: $140$

Each exterior angle is $\dfrac{360}{9} = 40^\circ\mathclose{,}$ and an interior and an exterior angle make a straight line: $180 - 40 = 140\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the exterior angle instead → $40$
• Gave the sum of all the interior angles → $1260$

Answer: $30$

The two legs are the base and the height. The other leg is $\sqrt{13^2 - 5^2} = 12\mathclose{,}$ so the area is $\dfrac{1}{2} \times 5 \times 12 = 30\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to halve → $60$
• Used the hypotenuse as the height → $\frac{65}{2}$

Answer: $\frac{13 \pi}{2}$

The arc is $\dfrac{90}{360} = \dfrac{1}{4}$ of the circumference $2\pi r\mathclose{:}$ $\dfrac{1}{4} \times 2\pi \times 13 = \frac{13 \pi}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the area of the sector instead → $\frac{169 \pi}{4}$
• Used the diameter in place of the radius → $13 \pi$

Answer: $53$

Corresponding angles at parallel lines are equal: $53^\circ\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the two angles to add up to $180^\circ$ → $127$

Answer: $100$

There are $1 + 3 + 5 = 9$ equal parts in $180^\circ\mathclose{,}$ so one part is $20^\circ$ and the largest angle is $5 \times 20 = 100\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the size of one part → $20$
• Shared out $360^\circ$ instead of $180^\circ$ → $200$

Answer: $10$

The exterior angles of any polygon add to $360^\circ\mathclose{,}$ so there are $\dfrac{360}{36} = 10$ of them — one per side.

Common mistakes, and the answer each one gives:

• Divided $180^\circ$ by the angle instead of $360^\circ$ → $5$

Answer: $29$

Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs. $20^2 + 21^2 = 841\mathclose{,}$ and $\sqrt{841} = 29\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the legs instead of their squares → $41$
• Subtracted the squares, which finds a LEG, not the hypotenuse → $\sqrt{41}$

Answer: $81 \pi$

Area $= \pi r^2 = \pi \times 9^2 = 81\pi\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the circumference formula $2\pi r$ → $18 \pi$
• Used the diameter in place of the radius → $324 \pi$

Answer: $58$

Vertically opposite angles are equal, so it is also $58^\circ\mathclose{.}$ (The two angles NEXT to it are $122^\circ$ each.)

Common mistakes, and the answer each one gives:

• Gave an angle next to it instead → $122$