Cuboid and cube
A matchbox is a cuboid and a dice is a cube. The paper to wrap the box is its surface area; the sand it holds is its volume.
A cuboid l × b × h has volume lbh, total surface area 2(lb + bh + hl) and lateral surface area 2h(l + b). A cube of side a has volume a³ and surface area 6a².
Formula: V = lbh; TSA = 2(lb + bh + hl); cube V = a³, TSA = 6a²
In the figure: A 5 × 4 × 3 cuboid: V = 60 and TSA = 2(20 + 12 + 15) = 94.
Rule 1: Volume and area — Volume in cubic units, area in square units.
- Easy: 6 × 5 × 2: V = 60.
- Medium: Cube side 4: V = 64, TSA = 96.
- Exam-type: Room 5 m × 4 m × 3 m: four walls = 2 × 3 × 9 = 54 m².
Rule 2: Work backwards — Use V = lbh to find a missing edge; cube root for a cube.
- Easy: V = 120, l = 6, b = 5: h = 4.
- Medium: Cube V = 125: side 5.
- Exam-type: 1 m³ = 1000 litres.
Doubling every edge makes the volume 8 times and the surface area 4 times.
Mixing up volume (cubic units) with area (square units).
"Volume cubes, area squares." A cube has 6 faces, each a², so 6a².
Practice questions
- 1A cube has how many faces, all equal squares?
- 2A cuboid measures 17 cm × 12 cm × 8 cm. What is its volume?
- 3The edge of a cube is 11 cm. Find its total surface area.
- 4A room is 7 m long, 19 m wide and 13 m high. What is the area of its four walls?
- 5The volume of a cuboid is 140 cm³. Its length is 7 cm and its width is 5 cm. Find its height.
- 6What is the volume of a cube of edge 14 cm?
- 7Find the length of the longest rod (the space diagonal) that fits inside a cuboid of 12 cm × 12 cm × 14 cm.
- 8How many cubes of edge 4 cm can be cut from a cuboid of 16 cm × 4 cm × 16 cm?
Show answers (try them yourself first)
- 16
- 21632 cm³
- 3726 cm²
- 4676 m²
- 54 cm
- 62744 cm³
- 722 cm
- 816
Cuboid and cube
In the app you can move this figure with your finger.
Open in the appCylinder
A tin can has a circular top and bottom with a curved side. Peel the label and it opens into a rectangle.
A cylinder of radius r and height h has curved surface area 2πrh (the rectangle 2πr × h), total surface area 2πr(r + h) and volume πr²h (base area × height).
Formula: CSA = 2πrh; TSA = 2πr(r + h); V = πr²h
In the figure: r = 7, h = 10: V = (22/7) × 49 × 10 = 1540 and CSA = 2 × (22/7) × 7 × 10 = 440.
Rule 1: Find area and volume — Use 22/7 when r is a multiple of 7.
- Easy: r = 7, h = 5: V = 770.
- Medium: r = 3.5, h = 10: CSA = 220.
- Exam-type: Open tank (no top): πr² + 2πrh.
Rule 2: Capacity problems — 1000 cm³ = 1 litre; convert units first.
- Easy: r = 7 cm, h = 10 cm: 1540 cm³ = 1.54 L.
- Medium: Volume 616, r = 7: h = 4.
- Exam-type: Doubling r makes V four times.
The curved surface unrolls into a rectangle; the circles are the two lids.
Using the diameter in place of the radius.
"Round base times height." Peel the label: a rectangle 2πr wide and h tall, so CSA = 2πrh.
Practice questions
- 1The curved surface area of a cylinder is
- 2The volume of a cylinder is
- 3A cylinder has radius 28 cm and height 28 cm. Find its curved surface area. (Use π = 22/7.)
- 4A cylinder has radius 21 cm and height 3 cm. Find its curved surface area. (Use π = 22/7.)
- 5A cylinder has radius 21 cm and height 4 cm. Find its curved surface area. (Use π = 22/7.)
- 6A cylinder has radius 21 cm and height 5 cm. Find its curved surface area. (Use π = 22/7.)
- 7Find the volume of a cylinder of radius 14 cm and height 12 cm. (Use π = 22/7.)
- 8Find the volume of a cylinder of radius 21 cm and height 11 cm. (Use π = 22/7.)
Show answers (try them yourself first)
- 12πrh
- 2πr²h
- 34928 cm²
- 4396 cm²
- 5528 cm²
- 6660 cm²
- 77392 cm³
- 815246 cm³
Cylinder
In the app you can move this figure with your finger.
Open in the appCone
An ice-cream cone or a clown's hat is a cone. Filling a cone and pouring it into a cylinder of the same base and height needs three cones.
A cone of radius r, height h and slant height l (l² = r² + h²) has volume ⅓πr²h, curved surface area πrl and total surface area πr(r + l).
Formula: l² = r² + h²; V = ⅓πr²h; CSA = πrl
In the figure: r = 3, h = 4 gives slant height l = 5, so V = ⅓π × 9 × 4 = 12π and CSA = π × 3 × 5 = 15π.
Rule 1: Find l first — Use Pythagoras to get the slant height.
- Easy: r = 6, h = 8: l = 10.
- Medium: r = 5, l = 13: h = 12.
- Exam-type: h = 24, l = 25: r = 7.
Rule 2: Volume and area — Remember the one-third in the volume, but not in the area.
- Easy: r = 7, h = 6: V = ⅓ × 22/7 × 49 × 6 = 308.
- Medium: r = 7, l = 25: CSA = 22/7 × 7 × 25 = 550.
- Exam-type: Cone V is one third of the cylinder with the same r and h.
The slant height l is always longer than h.
Using h (vertical height) in the curved surface area instead of l.
"Cone = one third of cylinder." Three cones fill one tin. For area use the slant side: π-r-l.
Practice questions
- 1A cone has base radius 21 cm and height 28 cm. Find its slant height.
- 2A cone has base radius 21 cm and height 72 cm. Find its slant height.
- 3A cone has base radius 14 cm and height 48 cm. Find its slant height.
- 4A cone has base radius 14 cm and slant height 50 cm. Find its height.
- 5A cone has base radius 7 cm and slant height 25 cm. Find its height.
- 6A cone has base radius 21 cm and slant height 75 cm. Find its height.
- 7A semicircular sheet of radius 35 cm is bent into an open cone, so that the radius of the sheet becomes the slant height. What is the base radius of the cone?
- 8A semicircular sheet of radius 14 cm is bent into an open cone, so that the radius of the sheet becomes the slant height. What is the base radius of the cone?
Show answers (try them yourself first)
- 135 cm
- 275 cm
- 350 cm
- 448 cm
- 524 cm
- 672 cm
- 717.5 cm
- 87 cm
Sphere and hemisphere
A football and a globe are spheres. Cut a globe along the equator and each half is a hemisphere, like a bowl.
A sphere of radius r has surface area 4πr² and volume (4/3)πr³. A hemisphere has curved surface area 2πr², total surface area 3πr² (adding the flat circle) and volume (2/3)πr³.
Formula: Sphere: 4πr², (4/3)πr³. Hemisphere: 2πr², 3πr², (2/3)πr³
In the figure: r = 7: sphere V = (4/3)(22/7)(343) = 4312/3 ≈ 1437.3 and SA = 4 × (22/7) × 49 = 616.
Rule 1: Sphere — Square r for area, cube r for volume.
- Easy: r = 7: SA = 616.
- Medium: r = 3: V = 36π.
- Exam-type: Diameter 14 ⇒ r = 7 before using formulas.
Rule 2: Hemisphere — Decide whether the flat top is included.
- Easy: r = 7 bowl (outside): CSA = 308.
- Medium: r = 7 solid: TSA = 462.
- Exam-type: r = 3: V = 18π.
Doubling r makes surface area 4 times and volume 8 times.
Forgetting the flat circle (πr²) in the total surface area of a hemisphere.
"Sphere: four pi r squared." Area has 4 and r², volume has 4/3 and r³. A hemisphere is half, plus the flat lid for 3πr².
Practice questions
- 1A solid metal sphere of radius 21 cm is melted and recast into small spheres of radius 7 cm. How many small spheres are made?
- 2A solid metal sphere of radius 28 cm is melted and recast into small spheres of radius 14 cm. How many small spheres are made?
- 3A solid metal sphere of radius 21 cm is melted and recast into small spheres of radius 10.5 cm. How many small spheres are made?
- 4A solid metal sphere of radius 35 cm is melted and recast into small spheres of radius 7 cm. How many small spheres are made?
- 5A solid metal sphere of radius 42 cm is melted and recast into small spheres of radius 21 cm. How many small spheres are made?
- 6The surface area of a sphere of radius r is
- 7The volume of a sphere of radius r is
- 8Find the surface area of a sphere of radius 28 cm. (π = 22/7)
Show answers (try them yourself first)
- 127
- 28
- 38
- 4125
- 58
- 64πr²
- 7⁴⁄₃πr³
- 89856 cm²
Sphere and hemisphere
In the app you can move this figure with your finger.
Open in the appPyramids and combined solids
A tent with a cone on a cylinder, or a pencil with a cone tip, is a combined solid: add the parts.
A pyramid has volume ⅓ × base area × height. For a combined solid, volume = sum of the parts; surface area = visible surfaces only (do not count the faces where parts are joined).
Formula: V(pyramid) = ⅓ Bh; combined V = V₁ + V₂
In the figure: Pyramid volume is a third of the prism with the same base and height.
Rule 1: Pyramids — Find the base area first, then take a third of base × height.
- Easy: Square base 6, h = 9: V = ⅓ × 36 × 9 = 108.
- Medium: Base area 30, h = 10: V = 100.
- Exam-type: Triangular base area 12, h = 5: V = 20.
Rule 2: Combined solids — Split into known solids, compute each and add (volume) or add visible parts (area).
- Easy: Cylinder r = 7, h = 10 + hemisphere r = 7: V = 1540 + 718.7 ≈ 2258.7.
- Medium: Cube 10 with a cube 2 cut out of a corner: V = 1000 − 8 = 992.
- Exam-type: Joined faces are not part of the outside area.
Always list the shapes and the surfaces that are hidden before calculating.
Adding the surface area of every part, including faces that are joined.
"Pyramid is a third; add the parts." Cone and pyramid both carry ⅓ in volume. Combined solids: add volumes, but skip the joined faces for area.
Practice questions
- 1A square pyramid has how many triangular faces?
- 2A square pyramid has a base of side 16 cm and height 6 cm. Find its volume.
- 3A square pyramid has a base of side 14 cm and height 24 cm. Find its volume.
- 4A square pyramid has a base of side 8 cm and slant height 5 cm. Find its lateral surface area.
- 5A square pyramid has a base of side 8 cm and slant height 5 cm. Find its total surface area.
- 6A square pyramid has a base of side 14 cm and slant height 25 cm. Find its total surface area.
- 7A square pyramid has a base of side 14 cm and height 24 cm. Find its slant height (the height of each triangular face).
- 8A square pyramid has a base of side 6 cm and height 4 cm. Find its slant height (the height of each triangular face).
Show answers (try them yourself first)
- 14
- 2512 cm³
- 31568 cm³
- 480 cm²
- 5144 cm²
- 6896 cm²
- 725 cm
- 85 cm