Key points about solving 2D and 3D problems using Pythagoras' theorem
Pythagoras' theorem states that for any right-angled triangle, the square of the The longest side in a right-angled triangle. is equal to the sum of the squares of the other two sides.
If the sides of the right-angled triangle are labelled π, π and π then Pythagoras' theorem can be written as the formula πΒ² + πΒ² = πΒ².
Pythagoras' theorem can be used to find the length of line segments and applied in two and three dimensions. Using Pythagoras' theorem in 3D is for Higher tier only.
Make sure you are confident in working with Pythagoras' theorem before solving these 2D and 3D problems.
Calculating the length of a line segment
A line segment is a part of a line which has two end points. When plotted on a set of axes, the points at the ends of the line segment can be expressed using The ordered pair of numbers that defines the position of a point..
Find the length of the line segment using Pythagoras' theorem by adding a vertical and horizontal line to form a right-angled triangle.
When just two points, π΄ and π΅, have been provided it is useful to sketch a diagram.
Follow the worked example below
GCSE exam-style questions
- Point πΆ has coordinates (1, 1). Point π· has coordinates (6, 5).
Use Pythagoras' theorem to find the length of the line segment πΆπ·.
Give the answer to one decimal place.
πΆπ· = 6Β·4 units to 1 d.p.
A right-angled triangle is formed by adding a horizontal line from πΆ and dropping a vertical line from π·.
The horizontal line measures 5 units. The vertical line measures 4 units.
Two sides of the triangle are known. Pythagoras's theorem is used to find the missing side.
- Label the sides of the triangle π, π and π.
Remember the hypotenuse, which is opposite the right-angle, should always be labelled π, The order for π and π does not matter.
Substitute the values into the equation πΒ² + πΒ² = πΒ².
πΒ² = 5Β² and πΒ² = 4Β².Calculate the value of the squares. 5Β² = 25 and 4Β² = 16.
Add the squares together to get 41. This is the value of πΒ².
The inverse of squaring is square rooting.
- Calculate the square root of 41 to find π. This gives the answer of π = 6.4031.
Therefore, rounding this to one decimal place, the length of πΆπ· is 6Β·4 units.
- Point πΈ has coordinates ( β 6, 6).
Point πΉ has coordinates (6, 1).
Use Pythagoras' theorem to find the length of the line segment πΈπΉ.
πΈπΉ = 13 units
A right-angled triangle is formed by adding a horizontal line from πΉ and dropping a vertical line from πΈ.
The horizontal line measures 12 units.
The vertical line measures 5 units.
Two sides of the triangle are known.
Pythagoras's theorem is used to find the missing side.
- Label the sides of the triangle π, π and π.
Remember the hypotenuse, which is opposite the right-angle, should always be labelled π, The order for π and π does not matter.
Substitute the values into the equation πΒ² + πΒ² = πΒ².
πΒ² = 12Β² and πΒ² = 5Β².Calculate the value of the squares. 12Β² = 144 and 5Β² = 25.
Add the squares together to get 169. This is the value of πΒ².
The inverse of squaring is square rooting.
- Calculate the square root of 169 to find π. This gives the answer of π = 13.
Therefore, the length of πΈπΉ is 13 units.
- Point π has coordinates (1, 4).
Point π has coordinates (6, 2).
Use Pythagoras' theorem to find the length of the line segment ππ.
Give the answer to one decimal place.
ππ = 5Β·4 units
- Sketch a set of axes with π at (1, 4) and π at (6, 2).
A right-angled triangle is formed by adding a horizontal line from π and dropping a vertical line from π.
The horizontal line measures 5 units.
The vertical line measures 2 units.
Two sides of the triangles are known.
Pythagoras's theorem is used to find the missing side.
- Label the sides of the triangle π, π and π.
Remember the hypotenuse, which is opposite the right-angle, should always be labelled π, The order for π and π does not matter.
Substitute the values into the equation πΒ² + πΒ² = πΒ².
πΒ² = 5Β² and πΒ² = 2Β².Calculate the value of the squares. 5Β² = 25 and 2Β² = 4.
Add the squares together to get 29. This is the value of πΒ².
The inverse of squaring is square rooting.
- Calculate the square root of 29 to find π. This gives the answer of π = 5Β·3851β¦
Therefore, rounding this to one decimal place, the length of ππ is 5Β·4 units.
Using Pythagoras' theorem in 2D
Pythagoras' theorem can be used to solve 2-dimensional problems which involve calculating a length in a right-angled triangle.
Identify these questions by recognising situations where a side of a right-angled triangle needs to be calculated and two sides are known.
It may be necessary to use Pythagoras' theorem more than once in a problem.
Follow the worked example below
GCSE exam-style questions
- A ladder of length 7 metres leans against a vertical wall.
The base of the ladder is 1Β·5 metres from the base of the wall.
Work out how far up the wall the ladder reaches.
Give the answer to 1 decimal place.
6Β·8 m to 1 d.p
The wall, ladder and floor form a right-angled triangle.
Two sides of the triangle are known. Pythagoras' theorem is used to find the missing side.
Label the sides of the triangle π, π and π.
Substitute the values into the equation πΒ² + πΒ² = πΒ².
πΒ² = 1Β·5Β² and πΒ² = 7Β². Replace πΒ² with βΒ², the variable used in the question.Calculate the value of the squares 1Β·5Β² = 2Β·25 and 7Β² = 49. This gives the equation βΒ² + 2Β·25 = 49.
Subtract 2Β·25 from both sides of the equation to work out the value of βΒ². This leads to the equation βΒ² = 46Β·75.
Calculate the square root of 46Β·75 to find π.
This gives the answer of β = 6Β·8373β¦
Therefore, when rounded to 1 decimal place, the height the ladder can reach is 6Β·8 m.
- A piece of wire is formed into a trapezium shape.
Calculate the total amount of wire needed to make the trapezium.
50 mm
The amount of wire needed to make the trapezium is equivalent to its perimeter.
The length of the unknown side is the hypotenuse of a right-angled triangle with a base measuring 12 mm and a height that is 15 β 10 = 5 mm.
Two sides of the triangle are known. Pythagoras's theorem is used to find the missing side.
- Label the sides of the triangle π, π and π.
Remember the hypotenuse, which is opposite the right angle, should always be labelled π. The order for π and π does not matter.
Substitute the values into the equation πΒ² + πΒ² = πΒ².
πΒ² = 12Β² and πΒ² = 5Β².Calculate the value of the squares. 12Β² = 144 and 5Β² = 25.
Add the squares together to get 169. This is the value of πΒ².
Calculate the square root of 169 to find π. This gives the answer of π = 13 mm.
Add the four sides to find the perimeter.
10 + 12 + 15 + 13 = 50
Quiz β Solving 2D problems using Pythagoras' theorem
Practise what you've learned about solving 2D problems using Pythagoras' theorem with this quiz.
Higher β Using Pythagoras' theorem in 3D
Use Pythagoras' theorem to solve 3-dimensional problems which involve calculating the length of a right-angled triangle.
For example, Pythagoras' theorem can be applied twice to find the distance between two opposite vertices in a cuboid.
When working in three-dimensions, it can be helpful to draw additional diagrams to help visualise the question.
Numbers expressed in questions or final answers may be written as surds.
A surd is an exact answer written as a square root.
To simplify a surd, identify a square number factor.
For example, β12 can be simplified because 4 is a square number factor of 12.
β12 = β4 Γ β3 = 2β3
Follow the worked example below
GCSE exam-style questions
- A cuboid measures 4 cm Γ 7 cm Γ 10 cm.
Calculate the length of πΈπΎ.
Give the answer to 1 decimal place.
πΈπΎ = 12Β·8 cm.
The length πΈπΎ is the hypotenuse of a right-angled triangle, πΈπΎπΊ, with a height 7 cm and unknown base πΈπΊ.
The length of πΈπΊ is the hypotenuse of a second right-angled triangle, πΈπΊπ», with height 4 cm and a base of
10 cm.
Two sides of the triangle are known.
Pythagoras' theorem is used to find the missing side.
For triangle, πΈπΊπ», Pythagoras' theorem states
πΊπ»Β² + πΈπ»Β² = πΈπΊΒ².
Substitute the values, πΊπ»Β² = 10Β² and πΈπ»Β² = 4Β², into the formula.
Calculate the value of the squares.10Β² = 100 and 4Β² = 16.
Add the squares together to get 116.
This is the value of πΈπΊΒ². πΈπΊ can be calculated but this is unnecessary for the next step.
For triangle, πΈπΎπΊ, Pythagoras' theorem states
πΈπΊΒ² + πΊπΎΒ² = πΈπΎΒ².
- Substitute the values into the formula.
From the previous calculation, πΈπΊΒ² = 116 and
πΊπΎΒ² = 7Β² = 49.
Add the values together to get 165. This is the value of πΈπΎΒ².
Calculate the square root of 165 to find πΈπΎ.
This gives the answer of πΈπΎ = 12Β·8453β¦
Therefore, when rounded to 1 decimal place, πΈπΎ = 12Β·8 cm.
- A cube has sides measuring 8 cm.
Calculate the length of π₯.
Express the answer in the form πβ3 where π is an integer.
8β3 cm
The length of π₯ can be calculated by applying Pythagoras' theorem twice.
The length of π₯ is the hypotenuse of a right-angled triangle, π΄πΆπΊ, with height 8 cm and unknown base π΄πΆ.
The length π΄πΆ is the hypotenuse of a second right-angled triangle, π΄πΆπ· , with height 8 cm and base 8 cm.
For triangle, π΄πΆπ·, Pythagoras' theorem states πΆπ·Β² + π΄π·Β² = π΄πΆΒ².
Substitute the values, πΆπ·Β² = 8Β² and π΄π·Β² = 8Β², into the formula.
Calculate the value of the squares. 8Β² = 64 and 8Β² = 64.
Add the squares together to get 128. This is the value of π΄πΆΒ².
For triangle, π΄πΆπΊ, Pythagoras' theorem states π΄πΆΒ² + πΆπΊΒ² = π΄πΊΒ².
Substitute the values into the formula. From the previous calculation π΄πΆΒ² = 128 and πΆπΊΒ² = 8Β² = 64.
Add the values together to get 192. This is the value of π₯Β².
Calculate the square root of 192 to find π₯.
Write this as a simplified surd, by finding the largest square number factor of 192.
The largest square number factor of 192 is 64.
π₯ = β192 = β64 Γ β3.
- The square root of 64 is 8.
Therefore, π₯ = 8β3.
Higher β Quiz β Using Pythagoras' theorem in 3D
Practise what you've learned about using Pythagoras' theorem in 3D with this quiz.
Now you've revised solving 2D and 3D problems with Pythagoras' theorem, why not look at calculations using the alternate segment theorem, tangents and chords?
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