Area and Volume

4 min readEasy10-question skill check

Area and volume questions are some of the most predictable points on the test — if you know the formulas, they're almost free. The catch is knowing which formula to grab and not mixing up perimeter with area.

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Area and Volume

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Area measures how much flat space a 2D shape covers, in square units. Perimeter is the distance around a shape — just add up all the side lengths. Volume measures how much space a 3D solid holds, in cubic units. The test gives you most volume formulas in a reference sheet, but the area ones you should just know cold.

Here are the essentials:

Rectangle: Area = length × width. Perimeter = 2(length + width).
Triangle: Area = ½ × base × height. The height is the perpendicular distance from the base to the opposite point — not necessarily a side. Perimeter is just the three sides added.
Circle: Area = πr², where r is the radius (distance from center to edge). Circumference (the perimeter of a circle) = 2πr. The diameter is 2r, all the way across.

For 3D solids:

Rectangular prism (box): Volume = length × width × height.
Cylinder: Volume = πr²h — that's the circle area times the height.
Sphere: Volume = (4/3)πr³.
Cone: Volume = (1/3)πr²h.

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Triangle area uses the base and the perpendicular height — here the two legs of a right triangle.

The single most important habit: match the units to the measurement. Perimeter and circumference are linear (just units, like cm). Area is squared (cm²). Volume is cubed (cm³). If a question gives you an area and asks for a side length, you'll be taking a square root somewhere.

Another key move: work backwards. If you know a rectangle's area is 42 and its width is 14, then length = 42 ÷ 14 = 3. Many test questions hand you the area or volume and ask for a missing dimension — just plug into the formula and solve.

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Cylinder volume = πr²h: the circle's area πr² multiplied by the height.

Finally, watch for the triangle height trap: the height must be perpendicular to the base. In a right triangle the two legs serve as base and height, but in other triangles the height may be a separate dashed line, and a slanted side is not the height.

Quick check

Check your understanding with a question from this topic:

A rectangle has length 3 and width 14. What is its area?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

A rectangle has a length of 9 and a width of 4. What is the area of the rectangle?

Example 2

A triangle has a base of 10 and a height of 6. What is the area of the triangle?

Example 3

A cylinder has a radius of 3 and a height of 7. What is the volume of the cylinder? (Use the formula V = πr²h and give your answer in terms of π.)

Common pitfalls

Confusing area with perimeter

Perimeter adds the sides (linear units); area multiplies dimensions (square units). Read whether the question wants the distance around or the space inside before choosing a formula.

Forgetting the ½ in triangle area

Triangle area is ½ × base × height, not base × height. Skipping the half doubles your answer — and the test always offers that doubled value as a wrong choice.

Using a slanted side as the height

The height of a triangle must be perpendicular to the base. A tilted side is longer than the true height, so plugging it in gives a too-big area. Look for the right-angle mark or a dashed perpendicular line.

Mixing up radius and diameter

Circle and cylinder formulas use the radius. If the problem gives the diameter, halve it first. Forgetting this makes circle area off by a factor of 4 (since the radius gets squared).

Key takeaways

Rectangle area = length × width; triangle area = ½ × base × height; circle area = πr².
Volume of a box = lwh; cylinder = πr²h — square the radius, then multiply by height.
Perimeter/circumference use plain units, area uses square units, volume uses cubic units.
When given the area or volume and asked for a side, plug into the formula and solve backwards.
Always use the radius (half the diameter) and keep the ½ in triangle area.

Watch & learn

Curated walkthroughs on Area and Volume. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

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