Home

Maths

Trigonometry

3D Pythagoras - Higher

3D Pythagoras - Higher

Videos

Summary

Exercises

Select Lesson

Exam Board

Select an option

Statistics

Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability

Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

Trigonometry

Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry

Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

Maps and scale drawings

Shapes and area

Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

Area and circumference of circles: Formulae

3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change

Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs

Coordinates and midpoints

Straight line graphs

Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

Quadratic graphs

Reciprocal and cubic graphs

Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

Distance-time graphs

Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra

Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Double brackets: Expanding and factorising

Double and triple brackets - Higher

Solving equations

Expressions, equations, formulae, functions and identities

Writing formulae and equations from word problems

Writing formulae and equations from diagrams

Rearranging formulae

Factorising quadratics

The quadratic formula - Higher

Complete the square - Higher

Algebraic fractions - Higher

Sequences

Finding the nth term

Solving inequalities

Inequalities on graphs - Higher

Iteration - Higher

Simultaneous equations: elimination and substitution

Non-linear simultaneous equations - Higher

Algebraic proof - Higher

Composite and inverse functions - Higher

Number

Types of numbers

Order of operations: BODMAS

Multiplying and dividing by powers of 10

Multiplying and dividing whole numbers

Multiplying and dividing decimals

Negative numbers: add, subtract, multiply, divide

Prime numbers and prime factorisation

Multiples, factors and prime factors

LCM and HCF

Fractions

Fractions, decimals and percentages

Writing recurring decimals as fractions

Rounding: Integers, decimal places, significant figures

Estimation

Error intervals

Upper and lower bounds - Higher

Powers and roots: Square and cube numbers

Laws of indices: multiply, divide, brackets

Index laws: negative and fractional indices - Higher

Surds: Simplify, add and subtract - Higher

Rationalising surds - Higher

Standard form calculations

Explainer Video

Tutor: Bilal

Summary

3D Pythagoras

In a nutshell

So far, you have used Pythagoras' theorem to find lengths of 222​ dimensional shapes. However, Pythagoras' theorem can also applied to find distances in 333​ dimensions.



Using Pythagoras' theorem in 333​ dimensions

It is best to show this with a sketch. Here is a cuboid ABCDEFGHABCDEFGHABCDEFGH​:



To find the distance AFAFAF​ (=d=d=d), first use Pythagoras' theorem on triangle ABHABHABH​ to find the distance AHAHAH​, and then use it again on triangle AFHAFHAFH​ to find the distance AFAFAF​.

​

Example 1

In the cuboid shown above, a=3cm,b=4cm,d=13cma=3cm,b=4cm,d=13cma=3cm,b=4cm,d=13cm. Find the value of ccc.


Find AHAHAH by focusing on the right-angled triangle ABHABHABH:

a2+b2=(AH)2a^2+b^2=(AH)^2a2+b2=(AH)2​​​​

​32+42=(AH)23^2+4^2=(AH)^232+42=(AH)2​​

​(AH)2=25(AH)^2=25(AH)2=25​​

​AH=25=5cmAH=\sqrt{25}=5cmAH=25​=5cm​​


Then, find ccc by focusing on the triangle AFHAFHAFH:

​(AH)2+c2=d2(AH)^2+c^2=d^2(AH)2+c2=d2​​

​52+c2=1325^2+c^2=13^252+c2=132​​

​​​25+c2=16925+c^2=16925+c2=169​​

​c2=144c^2=144c2=144​​

c=144=12cmc=\sqrt{144}=12cmc=144​=12cm​​​​​​

​c=12cm‾\underline{c=12cm}c=12cm​​​

​​​

​​

Other shapes

This formula can be applied to other 3D shapes, such as cubes and pyramids.


Example 2

A cube ABCDEFGHABCDEFGHABCDEFGH​ has side lengths 2cm2cm2cm​. What is the exact length of BEBEBE​ shown in the diagram?


Maths; Trigonometry; KS4 Year 10; 3D Pythagoras - Higher


The length BEBEBE can be found by using Pythagoras on the triangle BEGBEGBEG. However, to find the length BGBGBG, first use Pythagoras on the triangle BAGBAGBAG:

(BA)2+(AG)2=(BG)2(BA)^2+(AG)^2=(BG)^2(BA)2+(AG)2=(BG)2​​


Since it's a cube, all sides are of length 2cm2cm2cm, so substitute this for the lengths BABABA and AGAGAG​:

​22+22=(BG)22^2+2^2=(BG)^222+22=(BG)2​​

​(BG)2=8(BG)^2=8(BG)2=8​​

​BG=8cmBG=\sqrt{8} cmBG=8​cm​​


Now, use Pythagoras on the triangle BEGBEGBEG to find BGBGBG:

(BG)2+(EG)2=(BE)2(BG)^2+(EG)^2=(BE)^2(BG)2+(EG)2=(BE)2​​

(8)2+22=(BE)2(\sqrt{8})^2+2^2=(BE)^2(8​)2+22=(BE)2​​

8+4=(BE)28+4=(BE)^28+4=(BE)2​​

​(BE)2=12(BE)^2=12(BE)2=12​​

​​BE=12=23cmBE=\sqrt{12}=2\sqrt{3}cmBE=12​=23​cm​​

​BE=23cm‾\underline{BE=2\sqrt{3}cm}BE=23​cm​​​


Read more

Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
Length:
Triangles and regular polygons: Types and properties

Unit 1

Triangles and regular polygons: Types and properties

Pythagoras’ theorem

Unit 2

Pythagoras’ theorem

Jump Ahead

Score 80% to jump directly to the final unit.

This is the current lesson and goal (target) of the path

3D Pythagoras - Higher

Unit 3

3D Pythagoras - Higher

Final Test

Test reviewing all units to claim a reward planet.

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you know when to use Pythagoras' theorem in 3D?

When you have a 3d shape and have sides that travel across 3 different dimensions (width, height, and depth), you know you have to use Pythagoras' theorem twice.

How do you use Pythagoras' theorem in 3 dimensions?

Apply Pythagoras' theorem twice on two different triangles to find the 3 dimensional length.

©2020 - 2025 evulpo AG

Beta