Here is a solid square-based pyramid, VABCD. The base of the pyramid is a square of side 6 cm. The height of the pyramid is 4 cm. M is the midpoint of BC and VM = 5 cm. (a) Draw an accurate front elevation of the pyramid from the direction of the arrow. (b) Work out the total surface area of the pyramid.

GCSE Edexcel Maths Area & Perimeter: Here is a solid square-based pyr

Here is a solid square-based pyramid, VABCD - Edexcel - GCSE Maths - Question 5 - 2018 - Paper 1

Question 5

Here is a solid square-based pyramid, VABCD. The base of the pyramid is a square of side 6 cm. The height of the pyramid is 4 cm. M is the midpoint of BC and VM = 5... show full transcript

Worked Solution & Example Answer:Here is a solid square-based pyramid, VABCD - Edexcel - GCSE Maths - Question 5 - 2018 - Paper 1

Step 1

(a) Draw an accurate front elevation of the pyramid from the direction of the arrow.

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Answer

To draw the front elevation of the pyramid from the given direction:

Identify the Points: The base is square with points A, B, C, and D. Given that the side length is 6 cm, position A at the bottom left, B at (6,0), C at the top right (6,6), and D at the top left (0,6).
Position Point V: The height of the pyramid is 4 cm; from point M (which is the midpoint of BC, at (6,3)), extend vertically to point V directly above M, resulting in V located at (6, 3, 4).
Draw the Base: Draw a horizontal line from point A to B representing the base.
Connect the Points: Draw lines from V to points A, B, C, and D to complete the elevation.
Label the Diagram: Clearly label points A, B, C, D, and V in your drawing for clarity.

Step 2

(b) Work out the total surface area of the pyramid.

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Answer

To find the total surface area of the pyramid, we need to calculate the area of the base and the lateral surfaces:

Area of the Base: The base is a square with a side length of 6 cm, so its area (A_base) is given by:

$A_{base} = side^2 = 6^2 = 36 \, cm^2$
Area of the Triangular Faces: There are four triangular faces. The area of one triangle can be calculated using the formula:

$A_{triangle} = \frac{1}{2} \times base \times height$

For each triangle, the base is equal to the side of the base of the pyramid (6 cm), and the height can be calculated using the Pythagorean theorem for the right triangle formed:
- The height of the triangle is the slant height, which can be calculated as: $h = \sqrt{height^2 + \left(\frac{base}{2}\right)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, cm$
So, the area of one triangular face is:

$A_{triangle} = \frac{1}{2} \times 6 \times 5 = 15 \, cm^2$

Since there are four triangles:

$A_{triangles} = 4 \times A_{triangle} = 4 \times 15 = 60 \, cm^2$
Total Surface Area: The total surface area (A_total) is then:

$A_{total} = A_{base} + A_{triangles} = 36 + 60 = 96 \, cm^2$

Here is a solid square-based pyramid, VABCD - Edexcel - GCSE Maths - Question 5 - 2018 - Paper 1

Worked Solution & Example Answer:Here is a solid square-based pyramid, VABCD - Edexcel - GCSE Maths - Question 5 - 2018 - Paper 1

(a) Draw an accurate front elevation of the pyramid from the direction of the arrow.

(b) Work out the total surface area of the pyramid.

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