How to find perimeter, area and volume, even for odd shapes
TEAS 7 outline: Calculate geometric quantities
- Perimeter adds lengths, area is in square units, volume in cubic units.
- Rectangle area is L × W; triangle area is 1/2 × b × h.
- Circle area is π × r²; use the radius, not the diameter.
- Split odd shapes into simple shapes, then add or subtract their areas.
- Right triangles: a² + b² = c², where c is the longest side.
Key facts
- Rectangle: perimeter P = 2L + 2W; area A = L × W.
- Triangle: area A = 1/2 × b × h, where the height is measured perpendicular (at a right angle) to the base.
- Circle: diameter d = 2r; circumference C = π × d = 2 × π × r; area A = π × r²; use 3.14 for π unless told otherwise.
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- Volume of a rectangular solid is V = L × W × H; volume of a cylinder is V = π × r² × h.
- Perimeter is in linear units (ft), area in square units (ft²), volume in cubic units (ft³); 1 cm³ = 1 mL.
- Pythagorean theorem: in a right triangle, a² + b² = c², where c is the hypotenuse, the longest side, opposite the right angle.
- For an irregular shape, split it into rectangles, triangles and circle parts, then add (or subtract) their areas.
Quick check1 question · 2 more at the end
A round gauze pad has a radius of 3 cm. What is the area of one side of the pad? Use 3.14 for π.
Correct answer: D.
The area of a circle is A = π × r². Square the radius first: 3 × 3 = 9, then multiply by π: 3.14 × 9 = 28.26 cm². Area is always in square units; a result like 2 × π × r is the circumference, a length around the edge.
A This is 2 × 3.14 × 3, the circumference (distance around), not the area.
B This uses the diameter, 6 cm, as if it were the radius (3.14 × 6 × 6), making the area four times too big.
C This multiplies π by the radius without squaring it (3.14 × 3).
D Correct: A = π × r² = 3.14 × 3 × 3 = 28.26 cm².
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: Perimeter is the distance around. Area is the surface inside, in square units: rectangle L × W, triangle 1/2 × b × h, circle π × r². Volume is the space inside, in cubic units: box L × W × H, cylinder π × r² × h. Right triangles: a² + b² = c². Split odd shapes into simple ones.
1 Which formulas do I actually need?Perimeter measures the distance around, area the surface inside, and volume the space filled. 1 min
Perimeter measures the distance around, area the surface inside, and volume the space filled. Learn this short list and you can handle most shapes.
| Shape | Distance around | Area or volume |
|---|---|---|
| Rectangle | P = 2L + 2W | A = L × W |
| Triangle | add the 3 sides | A = 1/2 × b × h |
| Circle | C = π × d (or 2 × π × r) | A = π × r² |
| Box (rectangular solid) | n/a | V = L × W × H |
| Cylinder | n/a | V = π × r² × h |
The units tell you what you found: feet for perimeter, square feet (ft²) for area, cubic feet (ft³) for volume. Use 3.14 for π unless the question says otherwise.
2 How do I handle an irregular shape?Break an irregular shape into simple rectangles, then add or subtract. 1 min
Break an irregular shape into simple rectangles, then add or subtract. Sketch it and label every side before you calculate anything.
Worked example 1. A therapy room is an L shape: a 12 m by 9 m rectangle with a 4 m by 3 m corner cut out of it. Find its perimeter and floor area.
- Find the missing sides: the shortened top edge is 12 − 4 = 8 m, and the shortened side edge is 9 − 3 = 6 m.
- Walk around the outside and add all six edges: 12 + 9 + 8 + 3 + 4 + 6 = 42 m.
- Area of the full rectangle: 12 × 9 = 108 m².
- Subtract the cut-out corner: 4 × 3 = 12 m², so the area is 108 − 12 = 96 m².
Classic wrong turn: adding only the four labeled numbers (12 + 9 + 4 + 3 = 28 m). Every outside edge counts, including the ones you had to work out.
3 How do circles and cylinders work?Most circle mistakes come from mixing up the radius and the diameter. 1 min
Most circle mistakes come from mixing up the radius and the diameter. The radius is half the diameter. Circumference is π × d, area is π × r², and a cylinder's volume is its circle area times its height.
Worked example 2. A cylindrical water pitcher is 10 cm across and 12 cm tall inside. How many milliliters does it hold?
- Find the radius: 10 ÷ 2 = 5 cm.
- Area of the circular base: 3.14 × 5 × 5 = 78.5 cm².
- Multiply by the height: 78.5 × 12 = 942 cm³.
- Convert: 1 cm³ = 1 mL, so the pitcher holds 942 mL.
Classic wrong turn: using 10 as the radius, which gives 3,768 cm³. Doubling the radius multiplies the area, and the volume, by 4.
For a curved edge on an odd shape, remember that a semicircle's curve is half of π × d, and two equal semicircles make one full circle.
4 How does the Pythagorean theorem work?In any right triangle, the two legs squared add up to the hypotenuse squared. 1 min
In any right triangle, the two legs squared add up to the hypotenuse squared. The hypotenuse (c) is the longest side, across from the right angle.
Worked example 3. A banner string runs from the top of a 9 ft pole straight to a stake in the ground 12 ft from the pole's base. How long is the string?
- The pole and ground form the right angle, so the legs are 9 and 12 and the string is c.
- Square and add: 9 × 9 + 12 × 12 = 81 + 144 = 225.
- Take the square root: c = 15 ft (because 15 × 15 = 225).
Classic wrong turn: adding the legs (21 ft) or stopping at 225. Check: the hypotenuse must be longer than either leg but shorter than their sum.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: Use the radius, not the diameter, in π × r²; plugging in the diameter makes the area four times too big.
Watch out: Circumference is π × d, a length; area is π × r², in square units. Do not mix them up.
Watch out: The perimeter of an irregular shape includes every outside edge, including sides you must work out; inside dividing lines do not count.
Watch out: In a² + b² = c², square the legs before adding, then take the square root; the hypotenuse is not a + b.
Watch out: The height of a triangle meets the base at a right angle; a slanted side is not the height.
How to remember it
Write the units first, and the formula you need usually follows.
Rule to remember: around = add lengths (ft); inside = multiply two lengths (ft²); fill = multiply three lengths (ft³).
Memory hook: "Area wants r." Before using π × r², halve the diameter if that is what you were given.
Self-check:
- A circle has a diameter of 8 in. Find its circumference and area.
- Find the volume of a box 5 cm by 4 cm by 10 cm, in mL.
- A right triangle has legs 6 and 8. Find the hypotenuse.
Show answers
- C = 25.12 in and A = 50.24 in².
- Use 3.14 for π. Circumference: π × d = 3.14 × 8 = 25.12 in.
- Radius: 8 ÷ 2 = 4 in.
- Area: 3.14 × 4 × 4 = 3.14 × 16 = 50.24 in².
- 200 mL.
- Volume: 5 × 4 × 10 = 200 cm³.
- 1 cm³ = 1 mL, so the box holds 200 mL.
- 10.
- Square and add: 6 × 6 + 8 × 8 = 36 + 64 = 100.
- Take the square root: c = 10, because 10 × 10 = 100.
- Check: 10 is longer than 8 but shorter than 6 + 8 = 14.