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Geometry and Measurement - Surface Area and Volume of 3D Shapes (Cubes, Cuboids, Cylinders)

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A cube is a 3D shape where all faces are equal squares. If the edge length is ss, the volume measures the space inside (s×s×ss \times s \times s) and the total surface area is the sum of its six square faces (6×s26 \times s^2).

A 3D projection of a cube with equal side lengths s.
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A cuboid (rectangular prism) has six rectangular faces. Its dimensions are defined by length (ll), width (ww), and height (hh). Volume is V=lwhV = lwh and Total Surface Area is TSA=2(lw+wh+lh)TSA = 2(lw + wh + lh).

A cuboid showing length, width, and height dimensions.
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A cylinder consists of two congruent circular bases and a curved surface. The radius rr is the distance from the center to the edge of the base, and hh is the perpendicular distance between the bases.

A cylinder showing radius r and height h.
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Units of measurement are critical: Area is measured in square units (e.g., cm2cm^2) while Volume is measured in cubic units (e.g., cm3cm^3). Always ensure all dimensions are in the same unit before calculating.

📐Formulae

Volume of a Cube: V=s3V = s^3

Total Surface Area of a Cube: SA=6s2SA = 6s^2

Volume of a Cuboid: V=l×w×hV = l \times w \times h

Total Surface Area of a Cuboid: SA=2(lw+lh+wh)SA = 2(lw + lh + wh)

Volume of a Cylinder: V=πr2hV = \pi r^2 h

Curved Surface Area of a Cylinder: CSA=2πrhCSA = 2\pi rh

Total Surface Area of a Cylinder: SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi rh

Circumference of a Circle: C=2πrC = 2\pi r

💡Examples

Problem 1:

A rectangular water tank (cuboid) has a length of 88 m, a width of 55 m, and a height of 33 m. Calculate the volume of water it can hold and the total surface area of the tank's exterior.

Solution:

  1. Identify the dimensions: l=8l = 8, w=5w = 5, h=3h = 3.
  2. Calculate Volume: V=l×w×h=8×5×3=120 m3V = l \times w \times h = 8 \times 5 \times 3 = 120\text{ m}^3
  3. Calculate Surface Area: SA=2(lw+lh+wh)=2((8×5)+(8×3)+(5×3))SA = 2(lw + lh + wh) = 2((8 \times 5) + (8 \times 3) + (5 \times 3)) SA=2(40+24+15)=2(79)=158 m2SA = 2(40 + 24 + 15) = 2(79) = 158\text{ m}^2

Explanation:

To find the volume, we multiply all three dimensions together. To find the surface area, we calculate the area of the three pairs of identical rectangular faces and sum them up.

Problem 2:

A cylindrical soda can has a radius of 33 cm and a height of 1010 cm. Find the volume and the total surface area of the can. (Take π≈3.14\pi \approx 3.14)

Solution:

  1. Identify the dimensions: r=3r = 3, h=10h = 10.
  2. Calculate Volume: V=πr2h=3.14×32×10=3.14×9×10=282.6 cm3V = \pi r^2 h = 3.14 \times 3^2 \times 10 = 3.14 \times 9 \times 10 = 282.6\text{ cm}^3
  3. Calculate Total Surface Area: SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi rh SA=(2×3.14×32)+(2×3.14×3×10)SA = (2 \times 3.14 \times 3^2) + (2 \times 3.14 \times 3 \times 10) SA=(2×3.14×9)+(188.4)SA = (2 \times 3.14 \times 9) + (188.4) SA=56.52+188.4=244.92 cm2SA = 56.52 + 188.4 = 244.92\text{ cm}^2

Explanation:

The volume is found by multiplying the area of the circular base (πr2\pi r^2) by the height. The surface area is the sum of the two circular ends and the rectangular 'wrapped' side (circumference ×\times height).

Problem 3:

A wooden gift box is in the shape of a cube with an edge length of 1212 cm. Calculate the total surface area of the box and determine how many such boxes can be carved from a larger wooden block with a volume of 1036810368 cm3cm^3.

A cube with label 12 cm on the edge.

Solution:

  1. Find Total Surface Area (TSATSA): TSA=6s2TSA = 6s^2 TSA=6×(12)2=6×144=864 cm2TSA = 6 \times (12)^2 = 6 \times 144 = 864 \text{ cm}^2
  2. Find Volume (VV) of one box: V=s3V = s^3 V=123=1728 cm3V = 12^3 = 1728 \text{ cm}^3
  3. Find number of boxes: Number=Total VolumeVolume of one box\text{Number} = \frac{\text{Total Volume}}{\text{Volume of one box}} Number=103681728=6\text{Number} = \frac{10368}{1728} = 6 There are 6 boxes.

Explanation:

We first use the surface area formula for a cube to find the exterior area. To find how many fit in a larger volume, we calculate the volume of one cube and divide the total volume by this value.

Problem 4:

A cylindrical water pipe has an inner radius of 77 cm and a length (height) of 5050 cm. Calculate the Curved Surface Area (CSACSA) of the inside of the pipe and the volume of water it holds when full. (Use π=227\pi = \frac{22}{7})

Horizontal cylinder with radius 7 and height 50.

Solution:

  1. Calculate Curved Surface Area (CSACSA): CSA=2πrhCSA = 2\pi rh CSA=2×227×7×50CSA = 2 \times \frac{22}{7} \times 7 \times 50 CSA=2×22×50=2200 cm2CSA = 2 \times 22 \times 50 = 2200 \text{ cm}^2
  2. Calculate Volume (VV): V=πr2hV = \pi r^2 h V=227×72×50V = \frac{22}{7} \times 7^2 \times 50 V=227×49×50V = \frac{22}{7} \times 49 \times 50 V=22×7×50=7700 cm3V = 22 \times 7 \times 50 = 7700 \text{ cm}^3

Explanation:

The CSA of a cylinder only accounts for the side wall, not the top or bottom circles. The volume tells us the capacity of the pipe based on its cross-sectional area multiplied by its length.