Volume & Surface Area: Surface Area of Pyramid & Cone

Learn how to calculate the surface area of right pyramids, cones, and composite shapes. Interactive practice problems with step-by-step solutions.

Practice

📖 Learning Guide

1. Surface Area of a Right Pyramid

A right square pyramid has a flat square base and four triangular side faces. The surface area is the sum of the base area BB and the lateral area LL:
  • Base Area (BB): For a square base of side ss, B=s2B = s^2.
  • Lateral Area (LL): Composed of four identical triangles with base ss and slant height ll. The total lateral area is L=4×(12sl)=2slL = 4 \times \left(\frac{1}{2} s l\right) = 2 s l.
  • Slant Height (ll): If the perpendicular height hh is given instead of the slant height, calculate the slant height using the Pythagorean theorem on half the base side: l=h2+(s2)2l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}.

2. Surface Area of a Cone

A right circular cone consists of a flat circular base and a curved lateral surface. Its surface area is the sum of the base area BB and the lateral area LL:
  • Base Area (BB): For a circle of radius rr, B=πr2B = \pi r^2.
  • Lateral Area (LL): The unwrapped curved side forms a circular sector with area L=πrlL = \pi r l, where ll is the slant height.
  • Slant Height (ll): If height hh is given, find the slant height using the Pythagorean theorem: l=h2+r2l = \sqrt{h^2 + r^2}.

Surface Area of a Cone

A=πr2+πrlA = \pi r^2 + \pi r l
Where rr is the base radius and ll is the slant height.

💡 A circular sticker for the base (πr2\pi r^2) and a circular sector sticker for the curved side (πrl\pi r l) hovering outside.

lrBase Sticker (πr²)lSide Sticker (πrl)
Learning Topics

Frequently Asked Questions

What is volume?

Volume is the amount of three-dimensional space occupied by an object. It is measured in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).

How do you calculate the volume of a box?

The volume of a box (rectangular prism) is calculated by multiplying its length, width, and height: V = l × w × h.

What is the surface area of a shape?

Surface area is the total area of all the exterior faces of a 3D shape. It represents how many square units are needed to completely cover the outside of the shape without any overlaps.

How do you calculate the surface area of a right square pyramid?

The total surface area is: A=s2+2slA = s^2 + 2 s l, where ss is the side length of the square base and ll is the slant height. If perpendicular height hh is given, calculate l=h2+(s/2)2l = \sqrt{h^2 + (s/2)^2} first.

How do you calculate the surface area of a cone?

The total surface area is: A=πr2+πrlA = \pi r^2 + \pi r l, where rr is the base radius and ll is the slant height. If height hh is given, calculate l=h2+r2l = \sqrt{h^2 + r^2} first.

Surface Area of Pyramids & Cones — Practice & Guide | SealMath