Applications in two and three dimensions

Edexcel International GCSE Further Pure Mathematics· 10C· 25 min read

1. 2D Trigonometry Applications: Foundational Review★★☆☆☆⏱ 8 min

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Before working with 3D problems, we first review how to apply trigonometric rules to standard 2D problems, such as finding missing sides/angles in triangles, or solving navigation and surveying problems. The key skill here is to identify the correct triangle, label known values, and apply the sine, cosine, or tangent rule as appropriate. Remember that you must recall the sine rule and area of a triangle formula ( $\frac{1}{2} ab sin C$ ) for the exam, while the cosine rule is provided on the formula sheet.

📐 Worked Example

A surveyor stands 120m from the base of a vertical tower. The angle of elevation to the top of the tower is 32°. Calculate the height of the tower, to 3 significant figures.

1
Draw the right triangle formed by the surveyor, base of the tower, and top of the tower. Label adjacent side = 120m, angle = 32°, opposite side = h (height of tower).
2
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
3
Substitute known values: $tan (3 2^{\circ}) = \frac{h}{120}$
4
$h = 120 \times \tan(32^\circ) = 74.98 \approx 75.0\ \text{m}$

2. Calculating the angle between a line and a plane★★★☆☆⏱ 9 min

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For 3D problems, the first step is always to identify the relevant right triangle that corresponds to the angle you need to calculate. The angle between a line and a plane is the smallest angle between the line and its projection onto the plane. To find this angle: 1. Find the point where the line meets the plane (point of intersection). 2. Pick any other point on the line, and draw a perpendicular from that point down to the plane to find the projection point. 3. The right triangle is formed by the original line, its projection, and the perpendicular line. The angle between the original line and its projection is the required angle.

📘 Definition

Angle between a line and a plane

The smallest angle between a line and its orthogonal projection onto the plane, always between 0° and 90°.

📐 Worked Example

A cuboid has length 8cm, width 6cm, height 4cm. Find the angle between the space diagonal from the bottom-left front corner to the top-right back corner, and the base plane of the cuboid, to 1 decimal place.

1
First, calculate the length of the projection of the space diagonal onto the base plane: this is the base diagonal of the cuboid.
2
$\text{Base diagonal length} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10\ \text{cm}$
3
The right triangle for the angle has opposite side = height of cuboid (4cm), adjacent side = base diagonal (10cm).
4
$\tan(\theta) = \frac{4}{10} = 0.4$
5
$\theta = \arctan(0.4) \approx 21.8^\circ$

Exam tip:

Always state explicitly which two lines/planes form the angle you are calculating, to avoid losing marks for ambiguous working.

3. Calculating the angle between two planes (dihedral angle)★★★★☆⏱ 8 min

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The angle between two intersecting planes (called the dihedral angle) is the smallest angle between two lines, one on each plane, that are both perpendicular to the line where the two planes intersect. To calculate this angle: 1. Identify the line of intersection of the two planes. 2. Pick a point on this line of intersection, and draw a line perpendicular to the intersection line on each of the two planes. 3. The angle between these two perpendicular lines is the dihedral angle, which you can calculate using trigonometry on the resulting right triangle.

📘 Definition

Dihedral angle

The smallest angle between two intersecting planes, measured between two lines perpendicular to the planes' line of intersection, always between 0° and 90°.

📐 Worked Example

A square-based pyramid has a base side length of 10cm, and a perpendicular height of 12cm. Calculate the dihedral angle between one of the triangular faces and the base plane, to 1 decimal place.

1
Line of intersection between the triangular face and base is the base edge of the pyramid (10cm). The midpoint of this edge is 5cm from each end of the edge.
2
The perpendicular line on the base plane from the midpoint of the edge to the centre of the base is 5cm (since base is square, half the side length).
3
The perpendicular line on the triangular face is the slant height of the face, forming a right triangle with the perpendicular height of the pyramid (12cm) and the 5cm line on the base.
4
$\tan(\theta) = \frac{\text{perpendicular height of pyramid}}{5} = \frac{12}{5} = 2.4$
5
$\theta = \arctan(2.4) \approx 67.4^\circ$

4. Common Pitfalls

Wrong move:

Calculating the angle between a line and an edge of the plane, instead of the line's projection onto the plane.

Why:

This gives an incorrect angle that is larger than the true smallest line-plane angle, leading to lost marks.

Correct move:

Always find the orthogonal projection of the line onto the plane first, then calculate the angle between the original line and its projection.

Wrong move:

Failing to draw and label the extracted 2D triangle separately in your working.

Why:

Examiners cannot follow your reasoning for 3D problems without clear working, and you will lose partial credit even if your final answer is correct.

Correct move:

Draw the 2D triangle relevant to the angle you are calculating, label all known sides and angles, and reference it in your steps.

Wrong move:

Calculating the obtuse angle between two planes instead of the acute angle.

Why:

The specification requires the smallest possible angle between two planes, which is always between 0° and 90°.

Correct move:

If your calculation gives an angle greater than 90°, subtract it from 180° to get the correct acute dihedral angle.

Wrong move:

Using the wrong side lengths when setting up trigonometric ratios for 3D problems.

Why:

It is easy to mix up the height of a 3D shape with the slant height, leading to incorrect ratio calculations.

Correct move:

Label each side of your extracted 2D triangle with its role (opposite/adjacent/hypotenuse) relative to the angle you are calculating before substituting values.

Wrong move:

Rounding intermediate values too early in multi-step 3D problems.

Why:

Early rounding introduces error that can make your final answer fall outside the acceptable mark range.

Correct move:

Keep at least 4 significant figures for all intermediate calculations, and only round your final answer to the required number of significant figures.

5. Quick Reference Cheatsheet

Problem Type	Key Steps	Formula to Use
2D trigonometry problem	Label right/non-right triangle 2. Identify known sides/angles	SOHCAHTOA (right triangles); sine/cosine rule (non-right)
Angle between line and plane	Find line's projection onto plane 2. Extract right triangle with height/projection	$tan (θ) = \frac{perpendicular height}{projection length}$
Angle between two planes (dihedral)	Find line of intersection of planes 2. Draw perpendiculars on each plane to intersection line	$sin / cos / tan (θ)$ on right triangle formed by perpendiculars

6. Frequently Asked

Do I get marks for drawing a 2D triangle extracted from a 3D shape?

Yes! Examiners award partial credit for correctly identifying and labeling the relevant right triangle from a 3D figure, even if your final calculation is wrong. Always draw the 2D triangle separately and label all known sides/angles in your working.

What size angle should I calculate for line-plane or plane-plane problems?

You must always calculate the smallest possible angle between the two objects, so your answer will always be between 0° and 90° for these question types.

Going deeper

official_resourceEdexcel 4PM1 SpecificationOfficial specification for this topic (section 10C)

What's Next

Now that you have mastered 2D and 3D trigonometry applications, you are ready to move on to more advanced trigonometric topics in the Edexcel IGCSE Further Pure Maths syllabus. The next core topic is sine and cosine rules, which you will use to solve more complex non-right triangle problems in both 2D and 3D contexts. You will also build on these 3D geometry skills when solving problems involving mensuration of 3D shapes, including calculating surface areas and volumes of pyramids, prisms, and cones. Remember that these 3D trigonometry questions are almost always worth 4-6 marks in the exam, so practice extracting 2D triangles from different 3D figures (cuboids, pyramids, prisms) to build speed and accuracy. Make sure you memorize the steps for calculating line-plane and dihedral angles, as these are frequently tested question types.