13. Geometry

Angles, triangles, polygons, circles, Pythagoras and distances. Set for OMPT-C, OMPT-D and OMPT-G.

All OMPT worksheets

Questions

Sheet length

1

A right-angled triangle has a hypotenuse of $26$ and one leg of $10$ . How long is the other leg?

Answer:

2

Two angles of a triangle are $31^\circ$ and $110^\circ$ . How many degrees is the third?

Answer:

3

What is the sum of the interior angles of a polygon with $7$ sides, in degrees?

Answer:

4

A circle has a radius of $8$ . What is its circumference? Leave your answer in terms of $\pi$ .

Answer:

5

How far apart are the points $(-6, -2)$ and $(2, 13)$ ?

Answer:

6

A right-angled triangle has a hypotenuse of $25$ and one leg of $7$ . How long is the other leg?

Answer:

7

Two angles of a triangle are $76^\circ$ and $66^\circ$ . How many degrees is the third?

Answer:

8

What is the sum of the interior angles of a polygon with $9$ sides, in degrees?

Answer:

9

A circle has a radius of $9$ . What is its circumference? Leave your answer in terms of $\pi$ .

Answer:

10

How far apart are the points $(7, -6)$ and $(15, 9)$ ?

Answer:

11

A right-angled triangle has a hypotenuse of $10$ and one leg of $6$ . How long is the other leg?

Answer:

12

Two angles of a triangle are $20^\circ$ and $75^\circ$ . How many degrees is the third?

Answer:

13

What is the sum of the interior angles of a polygon with $16$ sides, in degrees?

Answer:

14

A circle has a radius of $10$ . What is its circumference? Leave your answer in terms of $\pi$ .

Answer:

15

How far apart are the points $(7, -8)$ and $(27, 13)$ ?

Answer:

16

A right-angled triangle has legs of length $10$ and $24$ . How long is its hypotenuse?

Answer:

17

Two angles of a triangle are $23^\circ$ and $113^\circ$ . How many degrees is the third?

Answer:

18

What is the sum of the interior angles of a polygon with $12$ sides, in degrees?

Answer:

19

A circle has a radius of $13$ . What is its area? Leave your answer in terms of $\pi$ .

Answer:

20

How far apart are the points $(8, -4)$ and $(18, 20)$ ?

Answer:

21

A right-angled triangle has legs of length $3$ and $4$ . How long is its hypotenuse?

Answer:

22

Two angles of a triangle are $27^\circ$ and $76^\circ$ . How many degrees is the third?

Answer:

23

What is the sum of the interior angles of a polygon with $8$ sides, in degrees?

Answer:

24

A circle has a radius of $6$ . What is its circumference? Leave your answer in terms of $\pi$ .

Answer:

25

How far apart are the points $(-7, 4)$ and $(-4, 8)$ ?

Answer:

26

A right-angled triangle has legs of length $20$ and $21$ . How long is its hypotenuse?

Answer:

27

Two angles of a triangle are $44^\circ$ and $69^\circ$ . How many degrees is the third?

Answer:

28

What is the sum of the interior angles of a polygon with $18$ sides, in degrees?

Answer:

29

A circle has a radius of $6$ . What is its area? Leave your answer in terms of $\pi$ .

Answer:

30

How far apart are the points $(-7, -5)$ and $(-4, -1)$ ?

Answer:

Answers

Every answer below was re-derived independently before this page was built.

1

Answer: $24$

Rearrange Pythagoras: the missing leg squared is $26^2 - 10^2 = 576$ , so the leg is $\sqrt{576} = 24$ .

Common mistakes, and the answer each one gives:

Subtracted the lengths instead of their squares → $16$
Added the squares, which is the rule for the HYPOTENUSE → $2 \sqrt{194}$

2

Answer: $39$

The three angles of any triangle add to $180^\circ$ , so the third is $180 - 31 - 110 = 39$ .

Common mistakes, and the answer each one gives:

Used $360^\circ$ , which is the angles round a POINT → $219$
Subtracted only one of the two given angles → $149$

3

Answer: $900$

A polygon with $7$ sides splits into $5$ triangles from one corner, and each triangle contributes $180^\circ$ : $(7 - 2) \times 180 = 900$ .

Common mistakes, and the answer each one gives:

Multiplied the number of sides by 180 without subtracting 2 → $1260$
Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

4

Answer: $16 \pi$

Circumference $= 2\pi r = 2 \times \pi \times 8 = 16\pi$ .

Common mistakes, and the answer each one gives:

Used the area formula $\pi r^2$ → $64 \pi$
Used the diameter as the radius → $32 \pi$

5

Answer: $17$

The horizontal gap is $8$ and the vertical gap is $15$ , so Pythagoras gives $\sqrt{8^2 + 15^2} = \sqrt{289} = 17$ .

Common mistakes, and the answer each one gives:

Added the two gaps instead of using Pythagoras → $23$
Forgot the square root → $289$

6

Answer: $24$

Rearrange Pythagoras: the missing leg squared is $25^2 - 7^2 = 576$ , so the leg is $\sqrt{576} = 24$ .

Common mistakes, and the answer each one gives:

Subtracted the lengths instead of their squares → $18$
Added the squares, which is the rule for the HYPOTENUSE → $\sqrt{674}$

7

Answer: $38$

The three angles of any triangle add to $180^\circ$ , so the third is $180 - 76 - 66 = 38$ .

Common mistakes, and the answer each one gives:

Used $360^\circ$ , which is the angles round a POINT → $218$
Subtracted only one of the two given angles → $104$

8

Answer: $1260$

A polygon with $9$ sides splits into $7$ triangles from one corner, and each triangle contributes $180^\circ$ : $(9 - 2) \times 180 = 1260$ .

Common mistakes, and the answer each one gives:

Multiplied the number of sides by 180 without subtracting 2 → $1620$
Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

9

Answer: $18 \pi$

Circumference $= 2\pi r = 2 \times \pi \times 9 = 18\pi$ .

Common mistakes, and the answer each one gives:

Used the area formula $\pi r^2$ → $81 \pi$
Used the diameter as the radius → $36 \pi$

10

Answer: $17$

The horizontal gap is $8$ and the vertical gap is $15$ , so Pythagoras gives $\sqrt{8^2 + 15^2} = \sqrt{289} = 17$ .

Common mistakes, and the answer each one gives:

Added the two gaps instead of using Pythagoras → $23$
Forgot the square root → $289$

11

Answer: $8$

Rearrange Pythagoras: the missing leg squared is $10^2 - 6^2 = 64$ , so the leg is $\sqrt{64} = 8$ .

Common mistakes, and the answer each one gives:

Subtracted the lengths instead of their squares → $4$
Added the squares, which is the rule for the HYPOTENUSE → $2 \sqrt{34}$

12

Answer: $85$

The three angles of any triangle add to $180^\circ$ , so the third is $180 - 20 - 75 = 85$ .

Common mistakes, and the answer each one gives:

Used $360^\circ$ , which is the angles round a POINT → $265$
Subtracted only one of the two given angles → $160$

13

Answer: $2520$

A polygon with $16$ sides splits into $14$ triangles from one corner, and each triangle contributes $180^\circ$ : $(16 - 2) \times 180 = 2520$ .

Common mistakes, and the answer each one gives:

Multiplied the number of sides by 180 without subtracting 2 → $2880$
Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

14

Answer: $20 \pi$

Circumference $= 2\pi r = 2 \times \pi \times 10 = 20\pi$ .

Common mistakes, and the answer each one gives:

Used the area formula $\pi r^2$ → $100 \pi$
Used the diameter as the radius → $40 \pi$

15

Answer: $29$

The horizontal gap is $20$ and the vertical gap is $21$ , so Pythagoras gives $\sqrt{20^2 + 21^2} = \sqrt{841} = 29$ .

Common mistakes, and the answer each one gives:

Added the two gaps instead of using Pythagoras → $41$
Forgot the square root → $841$

16

Answer: $26$

Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs. $10^2 + 24^2 = 676$ , and $\sqrt{676} = 26$ .

Common mistakes, and the answer each one gives:

Added the legs instead of their squares → $34$
Subtracted the squares, which finds a LEG not the hypotenuse → $2 \sqrt{119}$

17

Answer: $44$

The three angles of any triangle add to $180^\circ$ , so the third is $180 - 23 - 113 = 44$ .

Common mistakes, and the answer each one gives:

Used $360^\circ$ , which is the angles round a POINT → $224$
Subtracted only one of the two given angles → $157$

18

Answer: $1800$

A polygon with $12$ sides splits into $10$ triangles from one corner, and each triangle contributes $180^\circ$ : $(12 - 2) \times 180 = 1800$ .

Common mistakes, and the answer each one gives:

Multiplied the number of sides by 180 without subtracting 2 → $2160$
Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

19

Answer: $169 \pi$

Area $= \pi r^2 = \pi \times 13^2 = 169\pi$ .

Common mistakes, and the answer each one gives:

Used the circumference formula $2\pi r$ → $26 \pi$
Doubled the radius instead of squaring it → $26 \pi$

20

Answer: $26$

The horizontal gap is $10$ and the vertical gap is $24$ , so Pythagoras gives $\sqrt{10^2 + 24^2} = \sqrt{676} = 26$ .

Common mistakes, and the answer each one gives:

Added the two gaps instead of using Pythagoras → $34$
Forgot the square root → $676$

21

Answer: $5$

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

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 → $\sqrt{7}$

22

Answer: $77$

The three angles of any triangle add to $180^\circ$ , so the third is $180 - 27 - 76 = 77$ .

Common mistakes, and the answer each one gives:

Used $360^\circ$ , which is the angles round a POINT → $257$
Subtracted only one of the two given angles → $153$

23

Answer: $1080$

A polygon with $8$ sides splits into $6$ triangles from one corner, and each triangle contributes $180^\circ$ : $(8 - 2) \times 180 = 1080$ .

Common mistakes, and the answer each one gives:

Multiplied the number of sides by 180 without subtracting 2 → $1440$
Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

24

Answer: $12 \pi$

Circumference $= 2\pi r = 2 \times \pi \times 6 = 12\pi$ .

Common mistakes, and the answer each one gives:

Used the area formula $\pi r^2$ → $36 \pi$
Used the diameter as the radius → $24 \pi$

25

Answer: $5$

The horizontal gap is $3$ and the vertical gap is $4$ , so Pythagoras gives $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$ .

Common mistakes, and the answer each one gives:

Added the two gaps instead of using Pythagoras → $7$
Forgot the square root → $25$

26

Answer: $29$

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

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 → $\sqrt{41}$

27

Answer: $67$

The three angles of any triangle add to $180^\circ$ , so the third is $180 - 44 - 69 = 67$ .

Common mistakes, and the answer each one gives:

Used $360^\circ$ , which is the angles round a POINT → $247$
Subtracted only one of the two given angles → $136$

28

Answer: $2880$

A polygon with $18$ sides splits into $16$ triangles from one corner, and each triangle contributes $180^\circ$ : $(18 - 2) \times 180 = 2880$ .

Common mistakes, and the answer each one gives:

Multiplied the number of sides by 180 without subtracting 2 → $3240$
Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

29

Answer: $36 \pi$

Area $= \pi r^2 = \pi \times 6^2 = 36\pi$ .

Common mistakes, and the answer each one gives:

Used the circumference formula $2\pi r$ → $12 \pi$
Doubled the radius instead of squaring it → $12 \pi$

30

Answer: $5$

The horizontal gap is $3$ and the vertical gap is $4$ , so Pythagoras gives $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$ .

Common mistakes, and the answer each one gives:

Added the two gaps instead of using Pythagoras → $7$
Forgot the square root → $25$