Mensuration 3D: SSC CGL Quantitative Aptitude

Formulas (4)

• Volume and surface area of a cuboid and a cube · Cuboid and cube
  $$V = lbh \qquad S = 2(lb + bh + hl) \qquad V_{\text{cube}} = a^3 \qquad S_{\text{cube}} = 6a^2$$
• Walls of a room and the capacity of a tank · Four walls and capacity
  $$A_{\text{walls}} = 2(l + b)h \qquad 1\ \text{m}^3 = 1000\ \text{L} \qquad 1\ \text{L} = 1000\ \text{cm}^3$$
• Cutting a cuboid into cubes and joining cubes · Cutting and joining
  $$\text{Count} = \frac{V_{\text{big}}}{V_{\text{small}}} \qquad S_{n \text{ cubes in a row}} = (4n + 2)a^2$$
• The longest diagonal of a cuboid and a cube · Body diagonal
  $$d = \sqrt{l^2 + b^2 + h^2} \qquad d_{\text{cube}} = a\sqrt{3} \qquad d_{\text{face}} = a\sqrt{2}$$

Watch out for (8)

• Leaving out the 2 in the surface formula
  → Volume and surface area of a cuboid and a cube
• Edge from the perimeter of one face
  → Volume and surface area of a cuboid and a cube
• Taking the thickness off once
  → Walls of a room and the capacity of a tank
• Mixing metres and centimetres
  → Walls of a room and the capacity of a tank
• Adding the floor to the walls by habit
  → Walls of a room and the capacity of a tank
• Adding the surfaces of joined cubes
  → Cutting a cuboid into cubes and joining cubes
• One cut for every part
  → Cutting a cuboid into cubes and joining cubes
• Using the face diagonal for the longest rod
  → The longest diagonal of a cuboid and a cube

Formulas (4)

• Volume of a prism · Prism and two base areas
  $$V = \text{base area} \times h \qquad A_{\triangle} = \frac{\sqrt{3}}{4}a^2 \qquad A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}a^2$$
• Lateral surface area of a prism · Lateral and total surface of a prism
  $$\text{LSA} = \text{perimeter of base} \times h \qquad \text{TSA} = \text{LSA} + 2 \times \text{base area}$$
• A prism built in layers whose heights form an AP · Sum of the layer heights
  $$H = \frac{n}{2}(a + l) \qquad l = a + (n - 1)d \qquad V = \text{base area} \times H$$
• Volume and lateral surface of a pyramid · Pyramid
  $$V = \frac{1}{3} \times \text{base area} \times h \qquad \text{LSA} = \frac{1}{2} \times \text{perimeter} \times l$$

Watch out for (6)

• A hexagon is six triangles
  → Volume of a prism
• Square centimetres times metres
  → Volume of a prism
• Base area in place of perimeter
  → Lateral surface area of a prism
• Using one layer's height
  → A prism built in layers whose heights form an AP
• Forgetting the one third
  → Volume and lateral surface of a pyramid
• Slant height in the volume
  → Volume and lateral surface of a pyramid

Formulas (3)

• Volume and surface area of a cylinder · Cylinder
  $$V = \pi r^2 h \qquad \text{CSA} = 2\pi rh \qquad \text{TSA} = 2\pi r(r + h)$$
• Finding the radius from the curved surface or the circumference · Radius from the circumference, volume from the curved area
  $$r = \frac{C}{2\pi} \qquad V = \frac{r}{2} \times \text{CSA}$$
• Hollow cylinders, pipes and drilled holes · Pipe and drilled share
  $$V_{\text{metal}} = \pi(R^2 - r^2)h \qquad \text{share removed} = \frac{r^2}{R^2}$$

Watch out for (5)

• Using the diameter as the radius
  → Volume and surface area of a cylinder
• Two ends on an open vessel
  → Volume and surface area of a cylinder
• Rolling the sheet the wrong way
  → Finding the radius from the curved surface or the circumference
• Litres against cubic metres
  → Finding the radius from the curved surface or the circumference
• Subtracting the radii before squaring
  → Hollow cylinders, pipes and drilled holes

Formulas (3)

• Radius, height and slant height of a cone · Slant height
  $$l = \sqrt{r^2 + h^2}$$
• Volume of a cone · Volume of a cone
  $$V = \frac{1}{3}\pi r^2 h$$
• Curved and total surface area of a cone · Surface of a cone
  $$\text{CSA} = \pi rl \qquad \text{TSA} = \pi r(l + r)$$

Watch out for (6)

• Slant height used as the height
  → Radius, height and slant height of a cone
• Turning about the wrong side
  → Radius, height and slant height of a cone
• Leaving out the one third
  → Volume of a cone
• Slant height in the volume formula
  → Volume of a cone
• Adding a base to a tent
  → Curved and total surface area of a cone
• Height in the curved surface
  → Curved and total surface area of a cone

Formulas (4)

• Volume and surface area of a sphere and a hemisphere · Sphere and hemisphere
  $$V_{\text{sphere}} = \frac{4}{3}\pi r^3 \qquad S_{\text{sphere}} = 4\pi r^2 \qquad V_{\text{hemi}} = \frac{2}{3}\pi r^3 \qquad \text{CSA}_{\text{hemi}} = 2\pi r^2 \qquad \text{TSA}_{\text{hemi}} = 3\pi r^2$$
• Hollow spheres and hemispherical shells · Metal in a shell
  $$V_{\text{shell}} = \frac{4}{3}\pi(R^3 - r^3) \qquad V_{\text{hemi shell}} = \frac{2}{3}\pi(R^3 - r^3)$$
• When the volume is a multiple of the surface area · Volume over surface for a sphere
  $$\frac{V}{S} = \frac{r}{3}$$
• The largest sphere inside a cube, a box or a cone · Radius of the largest ball
  $$\rho_{\text{cube}} = \frac{a}{2} \qquad \rho_{\text{cuboid}} = \frac{\text{smallest edge}}{2} \qquad \rho_{\text{cone}} = \frac{rh}{r + l}$$

Watch out for (6)

• Curved or total surface of a hemisphere
  → Volume and surface area of a sphere and a hemisphere
• Diameter put in as the radius
  → Volume and surface area of a sphere and a hemisphere
• Cubing the thickness
  → Hollow spheres and hemispherical shells
• The wrong surface for a hemisphere
  → When the volume is a multiple of the surface area
• The longest edge limits the ball
  → The largest sphere inside a cube, a box or a cone
• Edge of the cube used as the radius
  → The largest sphere inside a cube, a box or a cone

Formulas (3)

• How a change in length changes area and volume · Scale factors
  $$\text{area} \times k^2 \qquad \text{volume} \times k^3 \qquad V_{\text{cone, cylinder}} \propto r^2 h$$
• Ratios of two cylinders or two cones · Two cylinders or two cones
  $$\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \qquad \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1 h_1}{r_2 h_2}$$
• Slicing a cone by a plane parallel to the base · Cut cone and frustum
  $$\frac{V_{\text{top}}}{V_{\text{whole}}} = \left(\frac{h}{H}\right)^3 \qquad V_{\text{frustum}} = \frac{1}{3}\pi h(R^2 + Rr + r^2)$$

Reference tables (1)

Watch out for (7)

• Adding the percentages
  → How a change in length changes area and volume
• Increased by 200% means three times
  → How a change in length changes area and volume
• New size or increase
  → How a change in length changes area and volume
• Squaring the height too
  → Ratios of two cylinders or two cones
• Measuring from the base
  → Slicing a cone by a plane parallel to the base
• Height ratio used as the volume ratio
  → Slicing a cone by a plane parallel to the base
• Cone against cylinder the wrong way round
  → Volume and surface formulas of the standard solids

Formulas (4)

• Melting and recasting keeps the volume the same · Volume is conserved
  $$V_{\text{before}} = V_{\text{after}} \qquad R^3 = r_1^3 + r_2^3 + r_3^3$$
• How many smaller solids, or how many containers · Number of pieces
  $$n = \frac{V_{\text{big}}}{V_{\text{small}}} \qquad n_{\text{same shape}} = \left(\frac{R}{r}\right)^3$$
• Surface area after recasting · Total surface after splitting into n equal copies
  $$S_{\text{after}} = \sqrt[3]{n} \times S_{\text{before}}$$
• Water displaced and the rise in level · Rise in water level
  $$\text{rise} = \frac{V_{\text{sunk}}}{\text{base area}}$$

Watch out for (6)

• Adding the radii
  → Melting and recasting keeps the volume the same
• Forgetting the one third in a cone
  → Melting and recasting keeps the volume the same
• Dividing the radii instead of the volumes
  → How many smaller solids, or how many containers
• Assuming the surface stays the same
  → Surface area after recasting
• Using the water already in the tank
  → Water displaced and the rise in level
• Dividing by the tank's volume
  → Water displaced and the rise in level