Math Worksheet for 15-Year-Olds: Mastering Measurements and Showing Workings

Instructions

For each problem below, calculate the required measurements. It is essential to show all of your workings clearly in the space provided. Marks are awarded for the correct method as well as the final answer. Use π ≈ 3.14 where necessary. Round your final answers to two decimal places if needed.

Section 1: 2D Shapes - Perimeter and Area

1. Calculate the perimeter and the area of the compound shape below. All corners are right angles.

             12 cm
         +---------+
         |         | 8 cm
         |         +---+
         |             | 5 cm
         +-------------+
               18 cm
           

   Workings:

   Perimeter: ________________ cm

   Area: ________________ cm²

2. A circle has a radius of 6 cm. Calculate its circumference and area.

   Workings:

   Circumference: ________________ cm

   Area: ________________ cm²

3. Find the length of the missing side (the hypotenuse) of the right-angled triangle below, and then calculate its perimeter and area.

                 |\
                 | \
              9 m|  \ ?
                 |   \
                 +----+
                  12 m
           

   Workings:

   Length of Hypotenuse: ________________ m

   Perimeter: ________________ m

   Area: ________________ m²

Section 2: 3D Shapes - Surface Area and Volume

4. A rectangular cardboard box has a length of 30 cm, a width of 15 cm, and a height of 20 cm. Calculate its volume and total surface area.

   Workings:

   Volume: ________________ cm³

   Surface Area: ________________ cm²

5. A cylindrical tin of soup has a radius of 4 cm and a height of 11 cm. Calculate its volume and total surface area.

   (Volume = πr²h)     (Surface Area = 2πrh + 2πr²)

   Workings:

   Volume: ________________ cm³

   Surface Area: ________________ cm²

Section 3: Application Problem

6. A bedroom is 5 metres long, 4 metres wide, and 2.5 metres high. You want to paint the four walls, but not the ceiling or the floor. One can of paint covers 20 m².

   1. Calculate the total area of the four walls.
   2. How many full cans of paint will you need to buy to give the walls one coat of paint?

   Workings:

   a) Total wall area: ________________ m²

   b) Cans of paint needed: ________________

Answer Key

Section 1: 2D Shapes - Perimeter and Area

1. Workings:

   Missing vertical side = 8 cm - 5 cm = 3 cm
   Missing horizontal side = 18 cm - 12 cm = 6 cm
   Perimeter = 12 + 8 + 6 + 5 + 18 + 3 = 52 cm
   Area can be split into two rectangles. Method 1:
   Area A (top) = 12 cm * 3 cm = 36 cm²
   Area B (bottom) = 18 cm * 5 cm = 90 cm²
   Total Area = 36 + 90 = 126 cm²
   (Method 2: (12*8) + (6*5) = 96 + 30 = 126 cm²)

   Perimeter: 52 cm

   Area: 126 cm²

2. Workings:

   Radius (r) = 6 cm
   Circumference = 2 * π * r = 2 * 3.14 * 6 = 37.68 cm
   Area = π * r² = 3.14 * 6² = 3.14 * 36 = 113.04 cm²

   Circumference: 37.68 cm

   Area: 113.04 cm²

3. Workings:

   Pythagoras' Theorem: a² + b² = c²
   9² + 12² = c²
   81 + 144 = c²
   225 = c²
   c = √225 = 15 m
   Perimeter = 9 + 12 + 15 = 36 m
   Area = (1/2) * base * height = 0.5 * 12 * 9 = 54 m²

   Length of Hypotenuse: 15 m

   Perimeter: 36 m

   Area: 54 m²

Section 2: 3D Shapes - Surface Area and Volume

4. Workings:

   Volume = length * width * height = 30 * 15 * 20 = 9000 cm³
   Surface Area = 2*(lw + lh + wh)
   Surface Area = 2 * ( (30*15) + (30*20) + (15*20) )
   Surface Area = 2 * ( 450 + 600 + 300 )
   Surface Area = 2 * (1350) = 2700 cm²

   Volume: 9000 cm³

   Surface Area: 2700 cm²

5. Workings:

   Volume = π * r² * h = 3.14 * 4² * 11 = 3.14 * 16 * 11 = 552.64 cm³
   Surface Area of curved side = 2 * π * r * h = 2 * 3.14 * 4 * 11 = 276.32 cm²
   Surface Area of two circular ends = 2 * π * r² = 2 * 3.14 * 4² = 2 * 3.14 * 16 = 100.48 cm²
   Total Surface Area = 276.32 + 100.48 = 376.80 cm²

   Volume: 552.64 cm³

   Surface Area: 376.80 cm²

Section 3: Application Problem

6. Workings:

   a) Find the area of the four walls:
   Area of two long walls = 2 * (length * height) = 2 * (5 * 2.5) = 2 * 12.5 = 25 m²
   Area of two wide walls = 2 * (width * height) = 2 * (4 * 2.5) = 2 * 10 = 20 m²
   Total wall area = 25 m² + 20 m² = 45 m²

   b) Find the number of cans needed:
   Total area to paint = 45 m²
   Coverage per can = 20 m²
   Number of cans = 45 / 20 = 2.25
   Since you cannot buy 0.25 of a can, you must buy a whole can. Therefore, 3 cans are needed.

   a) Total wall area: 45 m²

   b) Cans of paint needed: 3