Composite figures questions with answers

Questions

1. Question 1Area of a figure made of rectangles

   The diagram shows a figure made up of rectangles. All the corners are right angles. Find its area.

   Show the worked solution

   1. Step 1: Split the figure into two rectangles and find the missing side$\text{Missing side} = 18 - 7$$= 11$ cm
   2. Step 2: Add the areas of the two rectangles$\text{Area} = 7 \times 12 + 11 \times 6$$= 150$ cm²
2. Question 2Perimeter of a figure made of rectangles

   The diagram shows a figure made up of rectangles. All the corners are right angles. Find its perimeter.

   Show the worked solution

   1. Step 1: Find the missing sides: the gap's width, and its other side$\text{Gap} = 14 - 4 - 3$$= 7$ m$\text{Both sides of the gap are } 7$ m
   2. Step 2: Add all eight sides$\text{Perimeter} = 14 + 11 + 3 + 7 + 7 + 7 + 4 + 11$$= 64$ m
3. Question 3Rectangle and semicircle: area

   The figure is made up of a rectangle and a semicircle. Find its area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The semicircle's diameter is the side it stands on$r = \dfrac{12}{2}$$= 6$ m
   2. Step 2: Area = rectangle + half a circle$\text{Area} = 12 \times 8 + \dfrac{1}{2} \times \pi(6)^2$$= 152.5486\ldots$$= 153$ m² (3 s.f.)
4. Question 4Perimeters with arcs

   A running track is made up of two straight sections and two semicircles, as shown. Find the perimeter of the track, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The two semicircles make one full circle: its diameter is the width of the track$\text{Curved parts} = \pi d$$= \pi(12)$
   2. Step 2: Add the two straights (the dashed lines are inside the track)$\text{Perimeter} = 18 + 18 + \pi(12)$$= 73.6991\ldots$$= 73.7$ m (3 s.f.)
5. Question 5Shaded regions

   The diagram shows a square of side $6$ cm and a quarter circle with its centre at a corner of the square. Find the area of the shaded region, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Shaded area = square − quarter circle$\text{Area} = 6^2 - \dfrac{1}{4} \times \pi(6)^2$$= 7.7256\ldots$$= 7.73$ cm² (3 s.f.)
6. Question 6Rectangle and triangle

   The figure is made up of a rectangle and a triangle. The dashed line from the top of the triangle is perpendicular to the rectangle's side. Find the area of the figure.

   Show the worked solution

   1. Step 1: Rectangle: length × breadth$\text{Rectangle} = 22 \times 12$$= 264$ cm²
   2. Step 2: Triangle: its base is the rectangle's top; use the dashed height, not the slant side$\text{Triangle} = \dfrac{1}{2} \times 22 \times 8$$= 88$ cm²
   3. Step 3: Add the two areas$\text{Area} = 264 + 88$$= 352$ cm²
7. Question 7Two trapeziums: a missing length from the area

   The diagram shows a hexagon made of two identical trapeziums joined along their longer parallel sides (the dashed line). The parallel sides of each trapezium are $10$ cm and $16$ cm. The area of the hexagon is $78$ cm². Find the value of $h$.

   Show the worked solution

   1. Step 1: The two trapeziums are identical: each is half the hexagon$\text{One trapezium} = \dfrac{78}{2}$$\phantom{\text{One trapezium}} = 39$ cm²
   2. Step 2: Area of a trapezium = ½ × (sum of the parallel sides) × height: solve for h$\dfrac{1}{2} \times (10 + 16) \times h = 39$$13h = 39$$h = \dfrac{39}{13}$$h = 3$