Mensuration: perimeter, area, surface area, volume

Geometry: Mensuration 📐

Perimeter of 2‑D Shapes

The perimeter is the total length around a shape. Think of it as the distance you walk around a playground.

• Rectangle: $P = 2(l + w)$
• Square: $P = 4s$
• Triangle (any type): $P = a + b + c$
• Circle: $C = 2\pi r$ (circumference)

Area of 2‑D Shapes

Area tells you how many square tiles fit inside a shape. It’s measured in square units (e.g., cm², m²).

• Rectangle: $A = l \times w$
• Square: $A = s^2$
• Triangle: $A = \tfrac{1}{2} b h$
• Circle: $A = \pi r^2$
• Parallelogram: $A = b \times h$
• Trapezium: $A = \tfrac{1}{2}(a + b)h$

Surface Area of 3‑D Shapes

Surface area is the total area of all the faces of a 3‑D object. Imagine wrapping a gift: you need to know how much wrapping paper (surface area) you need.

• Cube: $SA = 6s^2$
• Rectangular Prism: $SA = 2(lw + lh + wh)$
• Sphere: $SA = 4\pi r^2$
• Cylinder: $SA = 2\pi r(h + r)$
• Right Circular Cone: $SA = \pi r(r + l)$ (where $l$ is slant height)

Volume of 3‑D Shapes

Volume measures how much space an object occupies. Think of it as how many water cups can fill the shape.

• Cube: $V = s^3$
• Rectangular Prism: $V = lwh$
• Sphere: $V = \tfrac{4}{3}\pi r^3$
• Cylinder: $V = \pi r^2 h$
• Right Circular Cone: $V = \tfrac{1}{3}\pi r^2 h$

Practice Example 🚀

Find the perimeter, area, surface area, and volume of a rectangular prism with length $l = 8$ cm, width $w = 5$ cm, and height $h = 3$ cm.

1. Perimeter of base (rectangle): $P_{\text{base}} = 2(l + w) = 2(8 + 5) = 26$ cm.
2. Area of base: $A_{\text{base}} = l \times w = 8 \times 5 = 40$ cm².
3. Surface area: $SA = 2(lw + lh + wh) = 2(40 + 24 + 15) = 2(79) = 158$ cm².
4. Volume: $V = lwh = 8 \times 5 \times 3 = 120$ cm³.

Formula Reference Table 📊