# 3D Trigonometry

Here we will learn about 3D trigonometry including how to combine your knowledge of Pythagoras’ Theorem, Trigonometric Ratios, The Sine Rule and The Cosine Rule and apply it to find missing angles and sides of triangles in 3-dimensional shapes.

There are also 3D trigonometry worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is 3D trigonometry?

3D trigonometry is an application of the trigonometric skills developed for 2 dimensional triangles.

To find missing sides or angles in 3 dimensional shapes, we need to be very clear which rules and formulae we need to use in order to find different angles and side lengths.

The flowchart below can help determine which function you need to use:

You may need to carry out this process several times in a question to fully answer what is being asked. You must be able to justify which rule or formulae you need to use.

Top Tip: Look out for common angles or common sides.

## How to use 3D trigonometry to find a missing angle or side length

In order to find a missing angle or side within a 3-dimensional shape:

1. Calculate the necessary missing angle or side of a triangle.
2. Sketch and label the second triangle using information from Step 1.
3. Calculate the missing angle or side of the final triangle.

## 3D trigonometry examples

### Example 1: missing side using trigonometry and Pythagoras’ theorem

The diagram shows a cuboid.

Calculate the length of the line AH . Write your answer in the form a\sqrt{b}\mathrm{cm} where a and b are integers.

1. Calculate the necessary missing angle or side of a triangle.

The line FH is shared between the two triangles EFH and AFH. We can find the length of FH by using simple trigonometric ratios.

Here we need to find the value of FH (the hypotenuse):

\begin{aligned} H&=\frac{O}{\sin(\theta)}\\\\ H&=\frac{18}{\sin(30)}\\\\ H&=36\mathrm{cm}\\\\ FH&=36\mathrm{cm} \end{aligned}

2Sketch and label the second triangle using information from step 1.

3Calculate the missing angle or side of the final triangle.

AFH is a right angle triangle, so we can use Pythagoras’ Theorem to find the value of x:

\begin{aligned} c^{2}&=a^{2}+b^{2}\\\\ x^{2}&=24^{2}+36^{2}\\\\ x^{2}&=1872\\\\ x&=12\sqrt{13}\mathrm{cm} \end{aligned}

### Example 2: finding length using trigonometric ratios

ABCDEF is an isosceles triangular prism. DF is 7cm, Angle DFE is 75o and angle ECF is 40o. Find the length of the line CE.

The triangle DEF is isosceles. We can split it in two and find the length of EF by using trigonometric ratios.

We need to find the value of x (the hypotenuse):

\begin{aligned} &H=\frac{A}{\cos(\theta)}\\\\ &x=\frac{3.5}{ \cos(75)}\\\\ &x=13.52296157\mathrm{cm} \end{aligned}

You must remember not to round your solution too early. Here we will continue to use the full decimal given for x = 13.52296157 .

Here we have a right angle triangle so we can use another trigonometric ratio to find the length CE:

\begin{aligned} H&=\frac{O}{\sin(\theta)}\\\\ CE&=\frac{13.52296157}{\sin(40)}\\\\ CE&=21.03799352…\\\\ CD&=21.34\mathrm{cm}\qquad(3.d.p) \end{aligned}

### Example 3:  missing angle including midpoint

ABCDEFGH is a cuboid with the following information:

• The front face of the cuboid is a square.
• The midpoint M lies half-way between E and F .
• The line AC is at 30o from the line AB .
• BCM is a triangle.
• AC = 52cm
• CE=26\sqrt{3}

Using the diagram below, calculate the size of angle BMC, to 2 decimal places.

The line BC is shared between the two triangles ABC and BCM. We can find the length of BC by using trigonometric ratios.

Here we need to find the value of BC (the opposite side to the angle \theta ):

\begin{aligned} O&=H\sin(\theta)\\\\ O&=52\sin(30)\\\\ O&=26\mathrm{cm}\\\\ BC&=26\mathrm{cm} \end{aligned}

As M is a midpoint along EF, this gives us an isosceles triangle BCM. As we know CE=26\sqrt(3) this is the vertical height of the triangle.

As we can split an isosceles triangle into two right-angle triangles, and we can use trigonometric ratios to find the angle \frac{\theta}{2} then multiply by 2 to find \theta :

\begin{aligned} \frac{\theta}{2}&=\tan^{-1}(\frac{O}{A})\\\\ \frac{\theta}{2}&=\tan^{-1}(\frac{13}{26\sqrt(3)})\\\\ \frac{\theta}{2}&=16.10211375…\\\\ \theta&=32.20\mathrm{cm}\qquad(2dp) \end{aligned}

### Example 4: missing angle using the cosine rule

ABCDEFGH is a cuboid. BH, FH and BF are straight lines that connect to make a triangle BFH. Using the information in the diagram, calculate the size of angle HFB.

Each side of the triangle BFH is the hypotenuse of one of the three faces of the cuboid. As we have all three dimensions of the cuboid, we can calculate each value (x, y, z) as follows:

Here we need to find the value of x (the hypotenuse):

\begin{aligned} x^{2}&=5^{2}+12^{2}\\\\ x&=\sqrt{25+144}\\\\ x&=13\mathrm{cm}\\\\ BF&=13\mathrm{cm}\\\\ \end{aligned}

Here we need to find the value of y (the hypotenuse):

\begin{aligned} y^{2}&=9^{2}+5^{2}\\\\ y&=\sqrt{81+25}\\\\ y&=\sqrt{106}\mathrm{cm}\\\\ BH&=\sqrt{106}\mathrm{cm} \end{aligned}

Here we need to find the value of z (the hypotenuse):

\begin{aligned} z^{2}&=9^{2}+12^{2}\\\\ z&=\sqrt{81+144}\\\\ z&=\sqrt{225}\mathrm{cm}\\\\ FH&=15\mathrm{cm} \end{aligned}

Keeping BH=\sqrt{106} , we now have the triangle BFH:

As we know the three side lengths of the triangle, we can use the cosine rule to find the missing angle \theta.

\begin{aligned} \cos(A)&=\frac{b^{2}+c^{2}-a^{2}}{2bc}\\\\ \cos(A)&=\frac{288}{390}\\\\ A&=cos^{-1}(\frac{288}{390})\\\\ A&=42.39947336…\\\\ A&=42.4^{\circ} (1.d.p) \end{aligned}

### Example 5: missing angle using the sine rule

The diagram shows a triangular prism.

• Angle DEC = 50º
• Angle EBC = 55º

The angle ϴ lies between the two lines CE and BE. Calculate the size of angle ϴ. Show all your working.

The line CE is shared between the two triangles CDE and BCE. We can find the length of CE by using trigonometric ratios.

Here we need to find the value of CE (the hypotenuse):

\begin{aligned} \\H&=\frac{O}{\sin(\theta)}\\ \\H&=\frac{25}{\sin(50)}\\ \\CE&=32.63518223…\mathrm{cm}\\ \end{aligned}

We know two sides and one of the two opposite angles so we need to use the sine rule to find the value for ϴ.

\begin{aligned} \\\sin(\theta)&=\frac{\sin(55)}{32.63518223…}\times{23}\\ \\\theta&=\sin^{-1}(0.5773063219)\\ \\\theta&=35.26130584…\\ \\\theta&=35.26^{\circ}\quad(2dp)\\ \end{aligned}

### Example 6: missing side using the sine rule

The diagram shows a cylinder:

• A, B and D are points on the circumference of the circles.
• C is the centre of the circle.
• ABC is a triangle
• AD is the diameter of the cylinder.

By calculating the size of angle ϴ, work out the height of the cylinder.

As we know two sides and one of the two opposite angles, we can use the sine rule to find the missing angle ϴ.

Here we need to find the value of ϴ:

\begin{aligned} \\\sin(\theta)&=\frac{\sin(32)}{7.8}\times{12.4}\\ \\\theta&=\sin^{-1}(0.8424357534)\\ \\\theta&=57.39823007…^{\circ}\\ \end{aligned}

As AD is the diameter of the circle, we can label AD as 2 × BC = 15.6cm.

As the angle CBD is 90º, we can calculate the angle ABD by 90 − 57.398…= 32.60176993º.

We now have enough information on the second triangle to calculate the height of the cylinder. Remember not to round too early.

To find the value of x, we use a simple trigonometric ratio.

\begin{aligned} \\A&=\frac{O}{\tan(\theta)}\\ \\x&=\frac{15.6}{\tan(32.60176993…)}\\ \\x&=24.39138008…\\ \\x&=24.39cm\quad(2dp)\\ \end{aligned}

### Common misconceptions

• Using Pythagoras’ Theorem instead of trigonometry

Using two sides of a non right-angle triangle to find the third side instead of using the cosine rule.

• Incorrect trigonometric ratio used

Incorrect labelling of any triangle can lead to the wrong trig function being used.

• Confusing the Sine Rule with the Cosine Rule

Misunderstanding when to use the sine rule or cosine rule to find a missing side or angle.

• Using the inverse trig function instead and inducing a mathematical error

If the inverse trig function is used instead of the standard trig function, the calculator may return a maths error as the solution does not exist.

3D trigonometry is part of our series of lessons to support revision on trigonometry. You may find it helpful to start with the main trigonometry lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

### Practice 3D trigonometry questions

1. ABCDEFGH is a cuboid. Calculate the length of DF to 2 decimal places.

9.04cm

10.33cm

7.53cm

15.82cm

Let’s first look at triangle FGH:

We want to know the length of FH which is the opposite.

###### \begin{aligned} O&=A \tan(\theta)\\\\ O&=17 \tan(28)\\\\ O&=9.039060338 \mathrm{cm} \end{aligned}

Now that we know the length of FH , we can consider the triangle DFH:

Since we know two sides and we want to calculate the third side, we can use Pythagoras Theorem:

 \begin{aligned}
c^{2}&=a^{2}+b^{2}\\\\
c^{2}&=5^{2}+9.039060338^{2}\\\\
c^{2}&=106.7046118\\\\
c&=\sqrt{106.7046118}\\\\
c&=10.33 \mathrm{cm}
\end{aligned} 

2. ABCDEF is a triangular prism. Calculate the angle DAE . Give your answer to 1 dp.

43.7^{\circ}

21.0^{\circ}

35.8^{\circ}

39.3^{\circ}

First we need to look at the triangle DEF:

We need to find the length of DE , which is the opposite.

\begin{aligned} O&=H \sin(\theta)\\\\ O&=9sin(62)\\\\ O&=7.946528336 \mathrm{cm} \end{aligned}

Now that we know the length of DE, we can consider the triangle ADE :

We want to find the angle DAE. We know O and A .

\begin{aligned} \tan(\theta)&=\frac{O}{A}\\\\ \tan(\theta)&=\frac{7.946528336}{11}\\\\ \theta&=\tan^{-1}(\frac{7.946528336}{11})\\\\ \theta&=35.8^{\circ} \end{aligned}

3. ABCDE is a square based pyramid. By finding the value of x , calculate the perimeter of the base of the pyramid, correct to 2 decimal places.

282.84m

400m

565.69m

70.71m

First we need to work out the value of x:

\begin{aligned} O&= H \sin(\theta)\\\\ O&=50 \sqrt{2} \sin(45)\\\\ O &=50 \mathrm{m} \end{aligned}

Now we can look at the base:

The length of the sides can be found using Pythagoras Theorem:

\begin{aligned} &c^2=a^2+b^2\\\\ &c^2=50^2+50^2\\\\ &c^2=5000\\\\ &c=70.71m \end{aligned}

Each side is 70.71m therefore the perimeter is:

4 \times 70.71=282.84 \mathrm{m}

4. Three satellites leave Earth on three different trajectories. 2 hours after launch, satellite A is 800km from Earth, satellite B is 500km from Earth and satellite C is 750km from Earth.

Use the cosine rule to calculate the size of angle CAB at this point in time.

76.60^{\circ}

38.22^{\circ}

32.01^{\circ}

51.78^{\circ}

AB, AC and BC can all be worked out using Pythagoras Theorem:

\begin{aligned} &AB^2=500^2+800^2\\ &AB^2=890000\\ &AB=943.3981132\\\\ &AC^2=750^2+800^2\\ &AC^2=1202500\\ &AC=1096.58561\\\\ &BC^2=500^2+750^2\\ &BC^2=812500\\ &BC=901.388189\\ \end{aligned}

We can then apply the cosine rule:

The length of the sides can be found using Pythagoras Theorem:

\begin{aligned} \cos(A)&=\frac{b^{2}+c^{2}-a^{2}}{2bc}\\\\ \cos(A)&=\frac{1096.58561^{2}+943.3981132^{2}-901.388189^{2}}{2 \times 1096.58562 \times 943.3981132}\\\\ \cos(A)&=0.6186459892\\\\ A&=\cos^{-1}(0.6186459892)\\\\ A&=51.87^{\circ} \end{aligned}

5. Given that GH=10cm, work out the size of the angle GEH . Give your answer to 1 dp.

38.7^{\circ}

38.5^{\circ}

51.3^{\circ}

36.4^{\circ}

First we need to calculate the length of EH. We can do this using Pythagoras Theorem:

\begin{aligned} a^{2}&=c^{2}-b^{2}\\\\ a^{2}&=20^{2}-12^{2}\\\\ a^{2}&=256\\\\ a&=16 \mathrm{cm} \end{aligned}

We can now look at the triangle EGH:

We can calculate angle GEH using the sine rule:

\begin{aligned} \frac{\sin(A)}{a} &= \frac{sin(B)}{b}\\\\ \frac{\sin(\theta)}{10} &= \frac{sin(85)}{16}\\\\ \sin(\theta)&= \frac{sin(85)}{16} \times 10\\\\ \sin(\theta)&= 0.6226216863\\\\ \theta&= \sin^{-1}(0.6226216863)\\\\ \theta&=38.5^{\circ} \end{aligned}

6. ABCDEF is a triangular prism. X, Y, and Z are midpoints on each edge of the prism and triangle XYZ is isosceles. Using this information and the diagram to help you, calculate the size of angle XYZ.

36.5^{\circ}

51.3^{\circ}

9.2^{\circ}

18.0^{\circ}

We need to calculate the length XZ using the triangle CXZ:

\begin{aligned} O&=A \tan(\theta)\\\\ O&=5 \tan(76)\\\\ O&=20.05390467 \mathrm{cm} \end{aligned}

We can now look at triangle XYZ:

Using the cosine rule:

\begin{aligned} \cos(A) &= \frac{b^{2}+c^{2}-a^{2}}{2bc}\\\\ \cos(\theta)&= \frac{32^{2}+32^{2}-20.05390467^{2}}{2 \times 32 \times 32}\\\\ \cos(\theta)&=0.8036332556\\\\ \theta&= \cos^{-1}(0.8036332556)\\\\ \theta&=36.5^{\circ} \end{aligned}

### 3D Trigonometry GCSE questions

1.  ABCDEFGH is a cuboid.

Calculate the angle between the diagonal DF and the base AEHD .

(4 marks)

DE^{2}=8^{2}+12^{2}

(1)

\begin{aligned} DE&=\sqrt{208}\\\\ DE&=14.4222\mathrm{cm} \end{aligned}

(1)

Triangle DEF:

\tan(\theta)=\frac{8}{14.4222}

(1)

\begin{aligned} \theta&=\tan^{-1}(\frac{8}{14.4222})\\\\ \theta&=29.0^{\circ} \end{aligned}

(1)

2.  ABCDEF is a triangular prism.

The cross-section of the prism is an isosceles triangle.

M is the midpoint of AC .

Calculate the length of EM .

(4 marks)

\tan(70)=\frac{x}{4.5}

(1)

\begin{aligned} 4.5\tan(70)&=x\\\\ 12.364&=x \end{aligned}

(1)

\begin{aligned} EM^{2}&=16^{2}+12.364^{2}\\\\ EM^{2}&=408.868 \end{aligned}

(1)

\begin{aligned} EM&=\sqrt{408.868}\\\\ EM&=20.2\mathrm{cm} \end{aligned}

(1)

3.  Find the size of the angle AFH .

(6 marks)

\begin{aligned} AF^{2}&=8^{2}+5^{2}\\\\ AF&=9.434 \end{aligned}

(1)

\begin{aligned} FH^{2}&=17^{2}+5^{2}\\\\ FH&=17.720 \end{aligned}

(1)

\begin{aligned} AH^{2}&=17^{2}+8^{2}\\\\ AH&=18.788 \end{aligned}

(1)

\cos(x)=\frac{9.434^{2}+17.720^{2}-18.788^{2}}{2\times9.434\times17.720}

(1)

\cos(x)=0.1096

(1)

\begin{aligned} x&=\cos^{-1}(0.1496)\\\\ x&=81.4^{\circ} \end{aligned}

(1)

## Learning checklist

You have now learned how to:

• Apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles (and, where possible, general triangles) in 2 (and 3) dimensional figures

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