Area and Volume of Prisms
This topic covers the fundamental formulae for calculating the area of common 2D shapes and the volume of 3D right prisms. These skills are essential building blocks often required in more complex, multi-step geometry problems.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The 'height' in any area formula must be the perpendicular height, measured at a 90-degree angle to the base, not a sloped side length.
- The volume of any right prism follows a single rule: find the area of the repeating cross-section and multiply it by the prism's length.
- A cuboid is simply a right prism with a rectangular cross-section, so its volume is (width × depth) × length.
- Be prepared for composite shapes, where you might need to add or subtract the areas of simpler shapes like triangles and rectangles.
Formulae
A = (1/2) × b × h To find the area of a triangle, where 'b' is the base and 'h' is the perpendicular height from the base to the opposite vertex.
A = b × h To find the area of a parallelogram, where 'b' is the base and 'h' is the perpendicular height between the base and the opposite parallel side.
A = (1/2) × (a + b) × h To find the area of a trapezium, where 'a' and 'b' are the lengths of the two parallel sides and 'h' is the perpendicular distance between them.
V = Acrosssection × L To find the volume of any right prism, where 'Acrosssection' is the area of its end face and 'L' is its length.
Definitions
- Right Prism
- A 3D solid with a consistent polygonal shape (the cross-section) along its entire length. The two end faces are parallel and identical.
- Cross-section
- The 2D shape exposed when a 3D object is sliced through. For a right prism, this shape is the same as its end face.
- Perpendicular Height
- The shortest distance from a base to the opposite side or vertex, forming a right angle with the base.
Worked example
A metal bar has a length of 50 cm. Its cross-section is a parallelogram with a base of 6 cm and a perpendicular height of 4 cm. The density of the metal is 8 g/cm3. What is the total mass of the bar in grams?
- 1
First, find the volume of the bar.
It's a right prism with a parallelogram cross-section.
- 2
Calculate the area of the cross-section:
A = base × height = 6 cm × 4 cm = 24 cm2 - 3
Calculate the volume of the prism:
V = Area × Length = 24 cm2 × 50 cm - 4
To calculate 24 × 50, you can do 24 × 100 / 2 = 2400 / 2 = 1200.
So, V = 1200 cm3 - 5
Now, calculate the mass using the density formula:
Mass = Density × Volume - 6 Mass = 8 g/cm3 × 1200 cm3
- 7
To calculate 8 × 1200, do 8 × 12 = 96, then add the two zeros.
Mass = 9600 g
Answer: 9600 g
Common mistakes
- ×Using a slanted side length instead of the perpendicular height. Exam questions often provide extra, irrelevant lengths to distract you.
- ×Forgetting the (1/2) factor for the area of triangles and trapezia. This is a frequent error leading to an answer that is exactly double the correct one.
- ×Making simple arithmetic mistakes under pressure, especially when multiplying multiple numbers together. Always double-check your calculations.
- ×Failing to convert units before calculating. If a prism's length is in meters and its cross-section area is in cm2, you must convert them to be consistent.
No-calculator tips
- ✓When calculating area of a trapezium or triangle, apply the '(1/2)' factor at the most convenient point. If the height or the sum of parallel sides is even, halve it first to work with smaller numbers.
- ✓To multiply by 50, it's often quicker to multiply by 100 and then halve the result. For example, 24 × 50 = 2400 / 2 = 1200.
- ✓Break down multiplications. To calculate 18 × 4, you can think of it as (20 × 4) - (2 × 4) = 80 - 8 = 72.