Area, Surface Area and Volume: Question 4

Worked solution

Part (a): Perimeter

The perimeter is the total distance all the way around the outside of the shape. The outline is made of six straight sides. Add them all up, going around in order:

$P = 10 + 6 + 4 + 8 + 6 + 14$

$P = 48\ \text{cm}.$

$\boxed{P = 48\ \text{cm}}$

Check: the two vertical sides on the right ($4\text{ cm}$ and $6\text{ cm}$) add up to $10\text{ cm}$, which matches the tall left side. The two horizontal sides across the top ($6\text{ cm}$ and $8\text{ cm}$) add up to $14\text{ cm}$, which matches the bottom edge. So the shape closes up correctly.

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Part (b): Area

An L-shape can be split into two rectangles. Draw a horizontal line across the shape at the height of the inner step (that is, $6\text{ cm}$ up from the bottom, since $10 - 4 = 6$). This gives:

Rectangle 1: the tall upright part (the left column of the L):

• width $= 6\text{ cm}$ (the top edge of the upright part)
• height $= 10\text{ cm}$ (the full tall left side)

$A_1 = 6 \times 10 = 60\ \text{cm}^2.$

Rectangle 2: the foot of the L (the part sticking out to the right):

• width $= 8\text{ cm}$ (the top of the foot)
• height $= 6\text{ cm}$ (the far right side)

$A_2 = 8 \times 6 = 48\ \text{cm}^2.$

Total area:

$A = A_1 + A_2 = 60 + 48 = 108\ \text{cm}^2.$

$\boxed{A = 108\ \text{cm}^2}$

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Alternative method (whole rectangle minus the missing corner)

You can also imagine the L filled in to make a large rectangle $14\text{ cm}$ wide and $10\text{ cm}$ tall, then subtract the rectangular piece that is missing from the top-right corner.

• Large rectangle: $14 \times 10 = 140\ \text{cm}^2$
• Missing corner: width $= 14 - 6 = 8\text{ cm}$, height $= 10 - 6 = 4\text{ cm}$, so $8 \times 4 = 32\ \text{cm}^2$

$A = 140 - 32 = 108\ \text{cm}^2.$

This agrees with the first method, which confirms the answer.