In this shape all the lengths are in centimetres. Work out (a) the perimeter, \_\_\_\_\_\_ $\text{cm}$ (b) the area. \_\_\_\_\_\_ $\text{cm}^2$

Mathematics (Syllabus D) 4024/11 — May/June 2013

Cambridge O-Level · Non-calculator · worked solutions for every part, with the mark scheme

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Topics Number · Mensuration · Geometry · Algebra and Graphs · Trigonometry · Statistics · +3 more

Q1MensurationFree sample

In this shape all the lengths are in centimetres.

Fig. 1

Work out

(a)

the perimeter,

______ $\text{cm}$

DifficultyEasy

Worked solution

Approach

To find the perimeter, determine the missing side lengths using the horizontal and vertical dimensions of the rectilinear shape, then sum all boundary edges.

Working

First, find the total horizontal width:

\text{Total width} = 5 + 25 = 30\,\text{cm}

The missing top-left horizontal edge is:

30 - 10 = 20\,\text{cm}

Next, find the vertical step height near the bottom left:

\text{Total height} = 20\,\text{cm}

\text{Height of the right section} = 20 - 5 = 15\,\text{cm}

\text{Bottom vertical step} = 15 - 12 = 3\,\text{cm}

Now, sum all the boundary side lengths:

\begin{aligned} \text{Perimeter} &= 20 + 20 + 5 + 10 + 12 + 25 + 3 + 5 \\ &= 100\,\text{cm} \end{aligned}

Answer

100\,\text{cm}

Final answer

100

Detailed explanation

Walkthrough

A rectilinear shape has all right angles, which means its boundary consists entirely of horizontal and vertical segments.
The total horizontal distance along the bottom is given by $5 + 25 = 30\,\text{cm}$ . The top edges must also add up to $30\,\text{cm}$ . Since the lower-step top edge is $10\,\text{cm}$ , the upper-step top edge is $30 - 10 = 20\,\text{cm}$ .
The total vertical height on the left is $20\,\text{cm}$ . Moving right, the top drops down by $5\,\text{cm}$ , leaving a height of $15\,\text{cm}$ above the main horizontal base. Since the rightmost vertical edge is $12\,\text{cm}$ , the vertical step up at the bottom left must be $15 - 12 = 3\,\text{cm}$ .
Summing all the horizontal segments ( $20 + 10 + 25 + 5 = 60\,\text{cm}$ ) and all the vertical segments ( $20 + 5 + 12 + 3 = 40\,\text{cm}$ ) gives a total perimeter of $60 + 40 = 100\,\text{cm}$ .

Key Takeaways

For any rectilinear shape, opposite parallel components must sum to the same total span.
The perimeter is the total distance around the outside edge of the shape.

Common Mistakes

Forgetting the inner step edges ( $5\,\text{cm}$ and $3\,\text{cm}$ ) when summing the perimeter.
Treating the shape as a simple rectangle of $20\,\text{cm} \times 30\,\text{cm}$ without accounting for the inward steps.

Things to Be Careful About

Ensure every side is accounted for by tracing around the perimeter systematically in one direction.

Techniques used

find missing side lengthscalculate the perimeter of a rectilinear polygon

(b)

the area.

______ $\text{cm}^2$

DifficultyMedium-Easy

Worked solution

Approach

Split the compound shape into three vertical rectangles, calculate the area of each, and add them together.

Working

Divide the shape into three vertical sections from left to right:

Left rectangle:
$\text{Width} = 5\,\text{cm}, \quad \text{Height} = 20\,\text{cm}$ $\text{Area}_1 = 5 \times 20 = 100\,\text{cm}^2$
Middle rectangle:
$\text{Width} = 20 - 5 = 15\,\text{cm}, \quad \text{Height} = 20 - 3 = 17\,\text{cm}$ $\text{Area}_2 = 15 \times 17 = 255\,\text{cm}^2$
Right rectangle:
$\text{Width} = 10\,\text{cm}, \quad \text{Height} = 12\,\text{cm}$ $\text{Area}_3 = 10 \times 12 = 120\,\text{cm}^2$

Sum the three areas:

\begin{aligned} \text{Total Area} &= 100 + 255 + 120 \\ &= 475\,\text{cm}^2 \end{aligned}

Answer

475\,\text{cm}^2

Final answer

475

Detailed explanation

Walkthrough

To find the total area of a compound shape made of perpendicular sides, split it into simpler rectangular sections.
Splitting vertically:
- The leftmost vertical strip has width $5\,\text{cm}$ and height $20\,\text{cm}$ , giving an area of $5 \times 20 = 100\,\text{cm}^2$ .
- The middle section has a width of $20 - 5 = 15\,\text{cm}$ and a height from the base line of $20 - 3 = 17\,\text{cm}$ , giving an area of $15 \times 17 = 255\,\text{cm}^2$ .
- The rightmost section has width $10\,\text{cm}$ and height $12\,\text{cm}$ , giving an area of $10 \times 12 = 120\,\text{cm}^2$ .
Alternatively, subtract missing corner cutouts from a bounding $20\,\text{cm} \times 30\,\text{cm}$ rectangle: $\text{Area} = (20 \times 30) - (5 \times 10) - (3 \times 25) = 600 - 50 - 75 = 475\,\text{cm}^2$
Both methods yield $475\,\text{cm}^2$ .

Key Takeaways

Compound areas can be found either by splitting into smaller rectangles or by subtracting missing regions from an enclosing rectangle.
The subtraction method is often faster and less prone to arithmetic mistakes.

Common Mistakes

Using incorrect dimensions for the sub-rectangles (e.g., using the full width of $25\,\text{cm}$ instead of subtracting the overlap).
Confusing perimeter and area calculations.

Things to Be Careful About

Ensure no overlapping regions are counted twice when adding separate rectangles.

Techniques used

split a compound shape into rectanglescalculate the area of rectanglessum partial areas

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