Perimeter, Area, Surface Area and Volume
Perimeter, Area, Surface Area and Volume covers calculating the perimeter and area of rectangles, triangles, parallelograms, trapezia and compound shapes, the circumference and area of circles, and the surface area and volume of 3D solids including cuboids and prisms.
Method
- Learn the area formulae by heart: rectangle = length x width, triangle = 1/2 x base x height, parallelogram = base x height, and trapezium = 1/2 x (a + b) x height, where a and b are the parallel sides.
- For circles, learn circumference = pi x diameter (or 2 x pi x radius) and area = pi x radius^2, using pi = 3.14 or a calculator's pi button as instructed.
- For a compound shape, split it into simple rectangles, triangles or other known shapes, calculate each area separately, then add them (or subtract, for a shape with a hole) to find the total.
- For perimeter, add the lengths of all the outside edges, working out any missing side lengths first from the given dimensions.
- For the volume of a prism, including a cuboid, multiply the area of the cross-section, the face that stays the same shape along its length, by the length of the prism.
- For the volume of a cylinder, multiply the area of the circular cross-section (pi x radius^2) by the height.
- For surface area, identify and calculate the area of every face of the solid separately, then add them together, taking care not to miss any hidden or repeated faces.
- Always check that every measurement is in the same unit before calculating, and give your final answer in the correct units (cm, cm^2 or cm^3).
Worked example
A cuboid-shaped water tank has a length of 40 cm, a width of 25 cm and a height of 30 cm. Calculate (a) the volume of the tank, and (b) its total surface area.
- Calculate the volume by multiplying all three dimensions: 40 x 25 x 30.
- Work this out in stages: 40 x 25 = 1000, then 1000 x 30 = 30000. This gives a volume of 30000 cm^3.
- For the surface area, find the area of each pair of opposite faces: top and bottom = 40 x 25 = 1000 cm^2 each (2000 in total), front and back = 40 x 30 = 1200 cm^2 each (2400 in total), and the two sides = 25 x 30 = 750 cm^2 each (1500 in total).
- Add all six faces together: 2000 + 2400 + 1500 = 5900. Final answer: volume = 30000 cm^3, surface area = 5900 cm^2.
Practice questions
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Q1Calculate the area of a triangle with base 12 cm and height 9 cm.Show answer
Answer: 54 cm^2 (1/2 x 12 x 9 = 54)
Q2Calculate the perimeter of a rectangle measuring 14 cm by 8.5 cm.Show answer
Answer: 45 cm (2 x (14 + 8.5) = 2 x 22.5 = 45)
Q3A trapezium has parallel sides of 6 cm and 10 cm, and a height of 5 cm. Calculate its area.Show answer
Answer: 40 cm^2 (1/2 x (6 + 10) x 5 = 1/2 x 16 x 5 = 40)
Q4Calculate the circumference of a circle with a radius of 9 cm. Use pi = 3.14.Show answer
Answer: 56.52 cm (circumference = 2 x 3.14 x 9 = 56.52)
Q5Calculate the area of a circle with a diameter of 20 cm. Use pi = 3.14.Show answer
Answer: 314 cm^2 (radius = 10 cm, area = 3.14 x 10^2 = 3.14 x 100 = 314)
Q6A cuboid measures 7 cm by 5 cm by 4 cm. Calculate its volume.Show answer
Answer: 140 cm^3 (7 x 5 x 4 = 140)
Q7An L-shaped room is made from a 6 m by 4 m rectangle with a 2 m by 3 m rectangle removed from one corner. Calculate the area of the room.Show answer
Answer: 18 m^2 (6 x 4 = 24, then subtract 2 x 3 = 6, giving 24 - 6 = 18)
Q8A triangular prism has a cross-sectional area of 15 cm^2 and a length of 12 cm. Calculate its volume.Show answer
Answer: 180 cm^3 (15 x 12 = 180)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A cylindrical water butt has a radius of 30 cm and a height of 80 cm. Using pi = 3.14, calculate the volume of the water butt in cm^3.
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A garden is in the shape of a rectangle measuring 12 m by 9 m, with a semicircular flower bed of diameter 6 m cut out from one of the 9 m sides (the flat edge of the semicircle lies along the 9 m side). Using pi = 3.14, calculate the remaining area of the garden after the flower bed is removed.
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A triangular prism has a triangular cross-section with base 8 cm and height 6 cm, and the prism is 15 cm long. Calculate the volume of the prism.
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Free printable worksheet
Want more practice on paper? Download the perimeter, area, surface area and volume worksheet pack - 9 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 18 of 13+ Maths Workbook, the whole course as one free printable PDF.
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