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Area, Perimeter,
and Volume

Master the essential formulas for measuring flat shapes and 3D solids: the building blocks of ISEE quantitative reasoning.

ISEE Middle Level

What Is Perimeter?

Perimeter is the total distance around the outside of a flat (2D) shape. Imagine walking along every edge and measuring how far you traveled: that total is the perimeter.

8 cm 8 cm 5 cm 5 cm

What Is Area?

Area measures how much surface a flat shape covers. Think of it as counting how many unit squares fit inside the shape. Area is always measured in square units (\( \text{cm}^2, \text{in}^2, \text{ft}^2 \)).

1 6 units 4 units
\( A = 6 \times 4 = 24 \text{ units}^2 \)

What Is Volume?

Volume measures how much space a 3D solid takes up. Imagine filling a box with unit cubes: the total number of cubes is the volume. Volume is measured in cubic units (\( \text{cm}^3, \text{in}^3, \text{ft}^3 \)).

5 cm 3 cm 4 cm

Perimeter vs. Area vs. Volume

Perimeter
Distance around a 2D shape. Measured in linear units (cm, in).
Area
Surface covered by a 2D shape. Measured in square units (\( \text{cm}^2, \text{in}^2 \)).
Volume
Space inside a 3D solid. Measured in cubic units (\( \text{cm}^3, \text{in}^3 \)).

Core Formulas: 2D Shapes

Shape Perimeter Area
Rectangle \( P = 2l + 2w \) \( A = l \times w \)
Square \( P = 4s \) \( A = s^2 \)
Triangle \( P = a + b + c \) \( A = \frac{1}{2}bh \)
Circle \( C = 2\pi r \) \( A = \pi r^2 \)

Core Formulas: 3D Solids

Solid Volume
Rectangular Prism \( V = l \times w \times h \)
Cube \( V = s^3 \)
Cylinder \( V = \pi r^2 h \)

Why \( A = \frac{1}{2}bh \) for Triangles?

b h
\( A_{\text{rectangle}} = b \times h \)
 Triangle = half the rectangle
\( A = \frac{1}{2}bh \)

Units Tell You What to Calculate

cm, ft, m
Linear units mean perimeter or circumference: you are measuring a distance.
\( \text{cm}^2, \text{ft}^2, \text{m}^2 \)
Square units mean area: you are measuring a flat surface.
\( \text{cm}^3, \text{ft}^3, \text{m}^3 \)
Cubic units mean volume: you are measuring a 3D space.

Step-by-Step Problem Solving

1
Identify the shape. Is it a rectangle, triangle, circle, or 3D solid?
2
Determine what to find. Check the units: linear = perimeter, square = area, cubic = volume.
3
Choose the correct formula and write it down before plugging in numbers.
4
Substitute and compute. Show your arithmetic step by step.
5
Include units in your answer. \( \text{cm}, \text{cm}^2, \text{or } \text{cm}^3 \): the exponent matters.

Example 1: Rectangle

A garden is 12 ft long and 7 ft wide. Find its perimeter and area.

12 ft 12 ft 7 ft 7 ft

Example 2: Triangle Area

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

10 cm 6 cm A B C

Example 3: Volume of a Box

A shipping box is 10 in long, 6 in wide, and 4 in tall. Find its volume.

10 in 4 in 6 in

Example 4: Circle Area and Circumference

A circle has a radius of 5 cm. Find its circumference and area. Use \( \pi \approx 3.14 \).

r = 5 cm
\( C = 2\pi r \)
\( = 2(3.14)(5) \)
\( = 31.4 \text{ cm} \)
\( A = \pi r^2 \)
\( = 78.5 \text{ cm}^2 \)

Common Mistakes to Avoid

✕ Forgetting the \( \frac{1}{2} \)
For triangle area, students compute \( b \times h \) instead of \( \frac{1}{2}bh \), getting double the correct answer.
✕ Mixing Up Radius and Diameter
If a problem gives the diameter, divide by 2 first. Using the diameter as \( r \) quadruples the area.
✕ Wrong Units
Writing \( \text{cm} \) for area instead of \( \text{cm}^2 \), or \( \text{cm}^2 \) for volume instead of \( \text{cm}^3 \). Units must match the dimension.

Formula Quick Reference

Shape / Solid Perimeter / Circumference Area Volume
Rectangle \( 2l + 2w \) \( lw \) —
Square \( 4s \) \( s^2 \) —
Triangle \( a + b + c \) \( \frac{1}{2}bh \) —
Circle \( 2\pi r \) \( \pi r^2 \) —
Rect. Prism — — \( lwh \)
Cube — — \( s^3 \)
Cylinder — — \( \pi r^2 h \)

Question 1 of 5

A rectangle has a length of 9 cm and a width of 4 cm. What is its perimeter?

Question 2 of 5

A triangle has a base of 14 in and a height of 8 in. What is its area?

Question 3 of 5

A cube has a side length of 5 cm. What is its volume?

Question 4 of 5

A circle has a diameter of 10 cm. What is its area? Use \( \pi \approx 3.14 \).

Question 5 of 5

A rectangular swimming pool is 12 m long, 5 m wide, and 2 m deep. How many cubic meters of water are needed to fill it?