Perimeter, area and volume

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Notes de leçon

Perimeter and area of 2D shapes

Perimeter is the distance around a shape; area is the space inside.
Rectangle: A = lw. Triangle: A = \frac{1}{2}bh. Parallelogram: A = bh. Trapezium: A = \frac{1}{2}(a + b)h.
Circle: circumference C = 2\π r = \π d; area A = \π r².
For compound shapes, split into simple parts and add or subtract.
Use perpendicular height for triangle, parallelogram and trapezium area. Perimeter adds all outside edges; include straight edges when finding a semicircle’s perimeter.

Trapezium: perpendicular height

Volume and surface area of 3D shapes

Prism: volume = cross-sectional area × length. Cuboid: V = lwh. Cylinder: V = \π r² h; curved surface area 2\π rh.
Cone: V = \frac{1}{3}\π r² h; curved surface area \π r l (slant height l).
Sphere: V = \frac{4}{3}\π r³; surface area 4\π r². Pyramid: V = \frac{1}{3} \times \text{base area} \times h.
For a closed cylinder, add both circular ends: total surface area = 2πr² + 2πrh. A right cone has total surface area πr² + πrl, with slant height l; an open solid omits its missing faces.
Cylinder and cone formulas here refer to right circular solids. The h in each volume formula is perpendicular height.

Cone: radius and perpendicular height

Units and conversions

Length: cm, m. Area: cm², m². Volume: cm³, m³. 1 m² = 10 000 cm²; 1 m³ = 1 000 000 cm³; 1 litre = 1000 cm³.
Always state the correct units.

Perimeter and area

Strategy

Draw and label diagrams, identify the shape, pick the formula, and substitute carefully, giving the answer in an appropriate form (in terms of \π or to 3 s.f.).

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Questions d'entraînement

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1.What is the area of a rectangle 5 cm by 8 cm?
Easy
- A13 cm²
- B26 cm²
- C40 cm²
- D80 cm²
2.What is the area of a circle with radius 4 cm? Leave in terms of \π.
Easy
- A8\π
- B16\π
- C4\π
- D64\π
3.What is the circumference of a circle with diameter 10 cm (in terms of \π)?
Easy
- A10\π cm
- B20\π cm
- C5\π cm
- D100\π cm
4.What is the area of a parallelogram with base 7 cm and perpendicular height 4 cm?
Easy
- A28 cm²
- B14 cm²
- C22 cm²
- D11 cm²
5.What is the area of a trapezium with parallel sides 6 cm and 10 cm and perpendicular height 4 cm?
Easy
- A32 cm²
- B64 cm²
- C24 cm²
- D40 cm²
6.A semicircle has diameter 10 cm. What is its area in terms of \π?
Easy
- A12.5\π cm²
- B25\π cm²
- C50\π cm²
- D5\π cm²
7.What is the volume of a cuboid 2 cm by 3 cm by 5 cm?
Easy
- A10 cm³
- B30 cm³
- C31 cm³
- D62 cm³
8.How many cm³ are in 1 litre?
Easy
- A10
- B100
- C1000
- D10 000

Plus de questions

9.What is the volume of a prism with cross-sectional area 15 cm² and length 8 cm?
Easy
- A23 cm³
- B120 cm³
- C60 cm³
- D240 cm³
10.What is the volume of a cylinder with radius 5 cm and height 12 cm? (3 s.f.)
Medium
- A942 cm³
- B314 cm³
- C188 cm³
- D3770 cm³
11.What is the volume of a cone with radius 3 cm and perpendicular height 4 cm in terms of \π?
Medium
- A12\π cm³
- B36\π cm³
- C6\π cm³
- D24\π cm³
12.What is the volume of a sphere with radius 6 cm in terms of \π?
Medium
- A288\π cm³
- B144\π cm³
- C864\π cm³
- D72\π cm³
13.What is the surface area of a sphere of radius 5 cm in terms of \π?
Medium
- A100\π cm²
- B25\π cm²
- C50\π cm²
- D500\π/3 cm²
14.What is the curved surface area of a cylinder with radius 4 cm and height 10 cm in terms of \π?
Medium
- A80\π cm²
- B160\π cm²
- C40\π cm²
- D32\π cm²
15.What is the volume of a square-based pyramid with base 6 cm by 6 cm and perpendicular height 5 cm?
Medium
- A60 cm³
- B180 cm³
- C90 cm³
- D30 cm³
16.Convert 3 m² to cm².
Medium
- A300 cm²
- B3000 cm²
- C30 000 cm²
- D300 000 cm²
17.A cylinder of volume 200\π cm³ has radius 5 cm. What is its height?
Medium
- A8 cm
- B40 cm
- C4 cm
- D10 cm
18.A sphere has volume 36\π cm³. What is its radius?
Medium
- A3 cm
- B6 cm
- C9 cm
- D4 cm
19.A right circular cone has radius 5 cm and slant height 13 cm. What is its perpendicular height?
Medium
- A12 cm
- B8 cm
- C18 cm
- D10 cm
20.The area of a circle is 49\π. What is its circumference in terms of \π?
Medium
- A14\π
- B7\π
- C49\π
- D98\π
21.A rectangle 10 cm by 6 cm has a circular hole of radius 2 cm removed. What area remains?
Medium
- A60 - 4\π
- B60 - 2\π
- C60 + 4\π
- D60 - 16\π
22.Which unit is used for volume?
Easy
- Acm
- Bcm²
- Ccm³
- Dm
23.The perimeter of a square is 20 cm. What is its area?
Easy
- A25 cm²
- B20 cm²
- C100 cm²
- D400 cm²
24.A tank is 2 m × 1.5 m × 1 m. How many litres does it hold?
Medium
- A3000
- B300
- C30 000
- D4.5
25.The area of a circle is \π r².
Easy
True or false?
26.The volume of a cone is \frac{1}{3}\π r² h.
Easy
True or false?
27.The area of a triangle is base times height.
Easy
True or false?
28.1 m² = 100 cm².
Easy
True or false?
29.The surface area of a sphere is 4\π r².
Medium
True or false?
30.Which are correct formulae?
Medium
- Acircle area \π r²
- Bcircumference 2\π r
- Ccylinder volume \π r² h
- Dsphere volume 4\π r²
- Etriangle area bh
31.Which are measured in cm²?
Medium
- Aarea of a rectangle
- Bvolume of a cube
- Csurface area of a sphere
- Dperimeter
- Earea of a circle
32.A cylinder has radius 4 cm and height 9 cm. Calculate its volume in terms of \π and its curved surface area in terms of \π.
Medium

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