Area and volume formulae
Derive and apply formulae for the area of triangles, parallelograms, and trapezia, and for the volume of cuboids and other prisms (including cylinders), connecting each formula to its geometric reasoning
What a learner can do afterwards
- Calculate the area of a trapezium using A = ½(a + b) × h and explain why the formula works
- Find the volume of a triangular prism by calculating cross-sectional area × length
- Derive the formula for the volume of a cylinder as π × r² × h by reasoning from the prism formula
1 · Read
You already know how to find the area of a triangle and a parallelogram. A trapezium is a four-sided shape with one pair of parallel sides that are different lengths. Its area formula is A = ½(a + b) × h, where a and b are the two parallel sides and h is the height between them.
Take a trapezium with parallel sides 6 cm and 10 cm, and a height of 4 cm. Split it into two triangles that share the same height. Their areas add up to (½ × 6 × 4) + (½ × 10 × 4), which is the same as ½ × (6 + 10) × 4 = 32 cm². That is why the formula averages the two parallel sides.
A prism is a solid with the same flat shape, called the cross-section, running through it from end to end. To find the volume of any prism, work out the area of the cross-section, then multiply it by the prism's length. Volume = cross-sectional area × length. A cuboid follows the same rule: its cross-section is a rectangle.
A cylinder is a prism whose cross-section is a circle. A circle's area is π × r², so that area does the same job as any other cross-section. Multiply it by the height and you get volume = π × r² × h. Only the radius is squared, and the height is multiplied in separately.
Trapezium area averages the parallel sides and multiplies by the height; any prism's volume, including a cylinder's, is cross-sectional area times length.
2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.