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Area, Perimeter,
and Volume

Master the essential formulas for 2D and 3D shapes and learn to solve ISEE-level problems involving area, perimeter, and volume.

ISEE Upper Level

What is Perimeter?

Perimeter is the total distance around the outside of a two-dimensional shape. Think of it as the length of fence needed to enclose a yard. You find it by adding up all the side lengths.

12 12 8 8
\(P = 12 + 8 + 12 + 8 = 40\)
 10 9 14
\(P = 10 + 9 + 14 = 33\)

What is Area?

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

5 3
\(A = 5 \times 3 = 15 \text{ sq units}\)
 r
\(A = \pi r^2\)

What is Volume?

Volume measures the amount of 3D space an object occupies. Think of filling a box with unit cubes. Volume is always in cubic units (\(\text{cm}^3\), \(\text{ft}^3\), \(\text{m}^3\)).

l h w
\(V = l \times w \times h\)
 r h
\(V = \pi r^2 h\)

Essential 2D Formulas

Rectangle
\(P = 2l + 2w\)
\(A = lw\)
Triangle
\(P = a + b + c\)
\(A = \tfrac{1}{2}bh\)
Circle
\(C = 2\pi r\)
\(A = \pi r^2\)
Trapezoid
\(A = \tfrac{1}{2}(b_1 + b_2)h\)
Parallelogram
\(A = bh\)

Essential 3D Formulas

Rect. Prism
\(V = lwh\)
\(SA = 2(lw + lh + wh)\)
Cylinder
\(V = \pi r^2 h\)
\(SA = 2\pi r^2 + 2\pi rh\)
Sphere
\(V = \tfrac{4}{3}\pi r^3\)
\(SA = 4\pi r^2\)
Cone
\(V = \tfrac{1}{3}\pi r^2 h\)
Pyramid
\(V = \tfrac{1}{3}Bh\)
(B = base area)

Perimeter vs Area vs Volume

Perimeter Area Volume
Measures Distance around Surface covered Space inside
Dimension 1D (length) 2D (surface) 3D (space)
Units cm, ft, m \(\text{cm}^2, \text{ft}^2, \text{m}^2\) \(\text{cm}^3, \text{ft}^3, \text{m}^3\)
Real-world Fencing a yard Painting a wall Filling a pool

How to Solve These Problems

1
Identify the shape. Is it a rectangle, triangle, circle, prism, cylinder, or composite shape?
2
Determine what to find. Is the question asking for perimeter, area, surface area, or volume? Check the units.
3
Select the right formula and write it down. Label the given values.
4
Substitute and compute. Plug values in, simplify carefully, and include the correct units.
5
Sanity-check. Does the answer make sense? Is it in the right unit type (linear, square, or cubic)?

Example 1: Rectangle Area & Perimeter

A rectangular garden has a length of 15 m and a width of 8 m. Find the perimeter and area.

15 m 8 m

Example 2: Circle Circumference & Area

A circular fountain has a radius of 6 ft. Find the circumference and area. Leave answers in terms of \(\pi\).

6 ft

Example 3: Composite Shape

A shape is formed by placing a semicircle (diameter 10 cm) on top of a rectangle (10 cm × 6 cm). Find the total area.

10 cm 6 cm d = 10, r = 5

Example 4: Cylinder Volume

A cylindrical water tank has a radius of 4 m and a height of 10 m. How many cubic meters of water can it hold?

4 m 10 m

Scaling: What Happens When Dimensions Change?

If every dimension of a shape is multiplied by a factor of \(k\), then:

Perimeter
Scales by \(k\)
Double sides → double perimeter
Area
Scales by \(k^2\)
Double sides → 4× area
Volume
Scales by \(k^3\)
Double sides → 8× volume

Finding Shaded Regions

Many ISEE problems show a shape with a region removed. The strategy is always the same:

10 r = 5
Area =
\(100 - 25\pi\)

Common Mistakes to Avoid

✕ Diameter vs Radius
Using the diameter where the formula needs the radius (or vice versa). Always check: \(r = \tfrac{d}{2}\).
✕ Wrong Units
Writing cm² for a volume answer, or cm for an area. Match the unit type to the measurement.
✕ Height vs Slant
Using the slant height instead of the perpendicular height in triangle and pyramid formulas.

Quick Reference Formula Table

Shape Perimeter / Circ. Area Volume
Rectangle \(2l + 2w\) \(lw\) —
Triangle \(a+b+c\) \(\tfrac{1}{2}bh\) —
Circle \(2\pi r\) \(\pi r^2\) —
Rect. Prism — \(2(lw+lh+wh)\) \(lwh\)
Cylinder — \(2\pi r^2+2\pi rh\) \(\pi r^2h\)
Sphere — \(4\pi r^2\) \(\tfrac{4}{3}\pi r^3\)

Question 1 of 5

A square has a perimeter of 36 cm. What is its area?

Question 2 of 5

A circle has a diameter of 14 inches. What is its area? (Use \(\pi \approx \tfrac{22}{7}\))

Question 3 of 5

A rectangular prism has dimensions 5 cm × 4 cm × 3 cm. What is its volume?

Question 4 of 5

If every edge of a cube is doubled, by what factor does its volume increase?

Question 5 of 5

A triangle and a rectangle both have a base of 12 cm and a height of 9 cm. How much larger is the rectangle's area than the triangle's area?