Geometry

Area & Volume Formulas

How much space does a shape cover? How much can a container hold? These are questions about area (2D) and volume (3D). Let’s explore the key formulas with interactive sliders so you can see how changing dimensions affects the result.

Part 1: Area of a Rectangle

The simplest area formula:

A = \text0 \times \text0

Length5

110

Width3

110

A = 5 \times 3

The area is the space enclosed. Double the length? Double the area. Double both dimensions? The area quadruples!

Try This

Try this: Set length = 4 and width = 4, then compare with length = 2 and width = 8. Both have area = 16, but very different shapes! Same area doesn’t mean same shape.

Part 2: Area of a Triangle

A triangle is half a rectangle:

A = \frac00 \times \text0 \times \text0

Base6

110

Height4

110

A = \frac00 \times 6 \times 4

Connection

Why the 1/2? Imagine a rectangle with the same base and height. Now cut it diagonally — each triangle is exactly half the rectangle. That’s where the 1/2 comes from.

Part 3: Area of a Circle

A = \pi r^2

The area of a circle depends on the radius squared. That means if you double the radius, the area gets four times bigger!

Radius (r)3

0.55

A = \pi \times 3^2

Try This

The power of squaring: A pizza with radius 6 inches has area pi * 36 = 113 square inches. A pizza with radius 12 inches has area pi * 144 = 452 square inches. Double the radius = 4x the pizza!

Part 4: Volume of a Rectangular Box

Moving to 3D! A box (rectangular prism) has:

V = \text0 \times \text0 \times \text0

Length4

Width3

Height5

V = 4 \times 3 \times 5

The graph shows how volume grows as you change the length (x-axis). The yellow line marks the current volume. Try changing width and height to see the slope change.

Part 5: Volume of a Cylinder

A cylinder is like a circular box:

V = \pi r^2 h

The base is a circle (area = pi * r^2), and you stack it up h units high.

Radius (r)3

0.55

Height (h)6

110

V = \pi \times 3^2 \times 6

The graph shows how volume increases linearly with height (for a fixed radius). The yellow line’s slope is pi * r^2 — the base area.

Part 6: Volume of a Sphere

V = \frac00\pi r^3

A sphere’s volume depends on the cube of the radius. Triple the radius and the volume increases by a factor of 27!

Radius (r)2

0.55

V = \frac00\pi \times 2^3

Let’s compare how different shapes’ volumes grow with radius:

Connection

Comparing shapes: For the same radius/side length, a sphere holds more than a cube at larger sizes because x^3 grows the same way but the sphere has the 4pi/3 multiplier (about 4.19). Nature loves spheres — they enclose the most volume for a given surface area. That’s why bubbles, planets, and water drops are round!

Part 7: Cones and Pyramids

A cone is like a cylinder that tapers to a point. Its volume is exactly one-third of the corresponding cylinder:

V_{\text0} = \frac00\pi r^2 h

Similarly, a pyramid is one-third of the prism with the same base:

V_{\text0} = \frac00 \times \text{base area} \times h

Radius3

0.55

Height6

110

V_{\text0} = \pi \times 3^2 \times 6

V_{\text0} = \frac00 \times \pi \times 3^2 \times 6

The cone is always exactly one-third of the cylinder with the same base and height. Change the radius slider and watch both lines change slope together — the ratio stays 3:1.

Challenge

Challenge:

A sphere has radius 3. A cylinder has radius 3 and height 6 (so the sphere fits perfectly inside). What fraction of the cylinder’s volume does the sphere occupy? (Hint: it’s a famous ratio!)
You have 500 cubic inches of clay. What’s the radius of the largest sphere you can make?
A cone and cylinder have the same base and height. How many cones of water would it take to fill the cylinder?

Wrapping Up

Shape	Area / Volume
Rectangle	A = l * w
Triangle	A = (1/2) * b * h
Circle	A = pi * r^2
Box	V = l * w * h
Cylinder	V = pi * r^2 * h
Sphere	V = (4/3) * pi * r^3
Cone	V = (1/3) * pi * r^2 * h

The key pattern: area formulas involve squaring a dimension, and volume formulas involve cubing one. That’s why doubling a dimension has such a dramatic effect on area and volume.

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