GCSE OCR Maths Vectors: An octahedron is formed from two

An octahedron is formed from two identical square based pyramids - OCR - GCSE Maths - Question 9 - 2019 - Paper 1

Question 9

An octahedron is formed from two identical square based pyramids. The square bases are stuck together as shown. The volume of the octahedron is 60 cm³. The length ... show full transcript

Worked Solution & Example Answer:An octahedron is formed from two identical square based pyramids - OCR - GCSE Maths - Question 9 - 2019 - Paper 1

Step 1

Determine the volume formula for the pyramid

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Answer

The volume V of a square pyramid is given by the formula:

$V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$

For a square base, the Base Area can be calculated as:

$\text{Base Area} = a^2$

where $a$ is the length of the side of the base.

Step 2

Calculate the base area

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Answer

Given that the length of the side of each pyramid's square base is 5 cm:

$\text{Base Area} = 5^2 = 25 \text{ cm}^2$

Step 3

Relate the volume of one pyramid to the octahedron's volume

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Answer

Since the octahedron consists of two identical pyramids, the volume of one pyramid is:

$V_{pyramid} = \frac{1}{2} \times 60 = 30 \text{ cm}^3$

Step 4

Set up the equation to find height h

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Answer

Now substituting the known values into the volume formula:

$30 = \frac{1}{3} \times 25 \times h$

Step 5

Solve for h

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Answer

To find the height $h$ , we can rearrange the equation:

$h = \frac{30 \times 3}{25} = \frac{90}{25} = 3.6 \text{ cm}$

An octahedron is formed from two identical square based pyramids - OCR - GCSE Maths - Question 9 - 2019 - Paper 1

Worked Solution & Example Answer:An octahedron is formed from two identical square based pyramids - OCR - GCSE Maths - Question 9 - 2019 - Paper 1

Determine the volume formula for the pyramid

Calculate the base area

Relate the volume of one pyramid to the octahedron's volume

Set up the equation to find height h

Solve for h

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