The diagram shows a pyramid ABCDE - OCR - GCSE Maths - Question 17 - 2021 - Paper 1

To find the slant height ED, we can use the Pythagorean theorem in triangle OED.

We know that:

• OE = 6.8 cm (height of the pyramid)
• OD = \frac{5.6}{2} = 2.8 cm (half the length of the base)

Using the Pythagorean theorem:

$$ED = \sqrt{OE^2 + OD^2} = \sqrt{6.8^2 + 2.8^2}$$

Calculating those values:

$$= \sqrt{46.24 + 7.84} = \sqrt{54.08} \approx 7.35 ext{ cm}$$

The area of one triangular face (e.g., ABE) can be calculated using the formula:

$$ext{Area}_{ ext{triangle}} = \frac{1}{2} \times ext{base} \times ext{height}$$

The base AB = 5.6 cm and height (slant height) ED = 7.35 cm:

$$ext{Area}_{ ext{ABE}} = \frac{1}{2} \times 5.6 \times 7.35 \approx 20.52 ext{ cm}^2$$

Since there are 4 triangular faces, the total lateral surface area can be calculated as:

$$ext{Total_{lateral}} = 4 \times ext{Area}_{ ext{triangle}} \approx 4 \times 20.52 \approx 82.08 ext{ cm}^2$$

Finally, the total surface area of the pyramid is the sum of the base area and the total lateral surface area:

$$ext{Total_{surface}} = ext{Area}_{ ext{base}} + ext{Total_{lateral}}$$

Substituting the values:

$$= 31.36 + 82.08 \approx 113.44 ext{ cm}^2$$