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. [Grid for drawing the front elevation] (b) Work out the total surface area of the pyramid.

GCSE Edexcel Maths Volume, Area & Surface Area: 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

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

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Answer

To create an accurate front elevation of the pyramid from the front view:

Base Representation: Start by drawing the square base of the pyramid, which has a side length of 6 cm. Make sure to represent the base symmetrically on your grid.
Vertex V: From the center of the base (which is 3 cm from each side), rise vertically to mark point V. Since the height of the pyramid is 4 cm, extend a line upwards from the center of the base to 4 cm to locate point V on your grid.
Connect Vertices: Connect point V with the corners A, B, C, and D of the base to show the triangular faces leading upwards.
Final Tweaks: Ensure all lines are straight and precise, representing the pyramid accurately on the grid provided.

Step 2

Work out the total surface area of the pyramid.

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Answer

To find the total surface area of the pyramid, follow these steps:

Base Area: The base of the pyramid is a square. Therefore, the area of the base A_base is calculated as:

$A_{base} = side^2 = 6^2 = 36 ext{ cm}^2$
Lateral Surface Area: The pyramid has 4 triangular faces. We'll calculate the area of one triangle first and then multiply by 4.
- Each triangular face has a base of 6 cm (the side of the square base) and a height that can be found using the Pythagorean theorem:
  - The slant height of the triangle can be found as:
  $h_{triangle} = ext{sqrt}(VM^2 - (base/2)^2) = ext{sqrt}(5^2 - 3^2) = ext{sqrt}(25 - 9) = ext{sqrt}(16) = 4 ext{ cm}$
- The area A_triangle of one triangular face is:
  
  $A_{triangle} = rac{1}{2} imes base imes height = rac{1}{2} imes 6 imes 4 = 12 ext{ cm}^2$
- Therefore, the total lateral surface area A_lateral is:
  
  $A_{lateral} = 4 imes A_{triangle} = 4 imes 12 = 48 ext{ cm}^2$
Total Surface Area: Finally, add the base area and the lateral surface area to find the total surface area A_total:

$A_{total} = A_{base} + A_{lateral} = 36 + 48 = 84 ext{ cm}^2$

Thus, the total surface area of the pyramid is 84 cm².

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

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

Work out the total surface area of the pyramid.

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