Circle, Coordinate, and Solid Geometry for IBA Admission

Volume of a rectangular solid with length $l$, breadth $b$, and height $h$

Formula: $V = l \times b \times h$

Example Question: How high will water rise if $12\,000\,\text{cm}^3$ of water is poured into a rectangular aquarium with length $50\,\text{cm}$ and breadth $30\,\text{cm}$?

Solution: $h = \frac{V}{l \times b} = \frac{12\,000}{50 \times 30} = 8\,\text{cm}$

Total surface area of a rectangular solid with length $l$, breadth $b$, and height $h$

Formula: $A = 2(lb + bh + lh)$

Example Question: What is the total surface area of a rectangular solid with length $5\,\text{cm}$, breadth $4\,\text{cm}$, and height $3\,\text{cm}$?

Solution: $A = 2(5 \times 4 + 4 \times 3 + 5 \times 3) = 2(20 + 12 + 15) = 94\,\text{cm}^2$

Space diagonal of a rectangular solid with length $l$, breadth $b$, and height $h$

Formula: $d = \sqrt{l^2 + b^2 + h^2}$

Example Question: What is the length of the longest pole that can fit in a rectangular room measuring $4\,\text{m}$ by $3\,\text{m}$ by $12\,\text{m}$?

Solution: $d = \sqrt{4^2 + 3^2 + 12^2} = \sqrt{16 + 9 + 144} = \sqrt{169} = 13\,\text{m}$

Volume of a cube with side length $a$

Formula: $V = a^3$

Example Question: What is the volume of a cube with side length $5\,\text{cm}$?

Solution: $V = 5^3 = 125\,\text{cm}^3$

Total surface area of a cube with side length $a$

Formula: $A = 6a^2$

Example Question: What is the total surface area of a cube having an edge length of $4\,\text{cm}$?

Solution: $A = 6(4^2) = 6 \times 16 = 96\,\text{cm}^2$

Main diagonal of a cube with side length $a$

Formula: $d = a\sqrt{3}$

Example Question: What is the space diagonal of a cube with side length $6\,\text{cm}$?

Solution: $d = 6\sqrt{3}\,\text{cm}$

Volume of a sphere with radius $r$

Formula: $V = \frac{4}{3}\pi r^3$

Example Question: What is the volume of a solid spherical toy with radius $r = 42\,\text{cm}$?

Solution: $V = \frac{4}{3}\pi(42)^3 = 98\,784\pi\,\text{cm}^3$

Surface area of a sphere with radius $r$

Formula: $A = 4\pi r^2$

Example Question: If the surface area of a solid sphere is $7\,056\pi\,\text{cm}^2$, what is its radius?

Solution: $4\pi r^2 = 7\,056\pi \implies r^2 = 1\,764 \implies r = 42\,\text{cm}$

Volume of a cylinder with base radius $r$ and height $h$

Formula: $V = \pi r^2 h$

Example Question: What is the volume of a cylinder with base radius $7\,\text{m}$ and height $6\,\text{m}$?

Solution: $V = \frac{22}{7} \times 7^2 \times 6 = 924\,\text{m}^3$

Curved surface area of a cylinder with radius $r$ and height $h$

Formula: $A = 2\pi rh$

Example Question: What is the curved surface area of a cylindrical pillar with radius $7\,\text{m}$ and height $6\,\text{m}$?

Solution: $A = 2 \times \frac{22}{7} \times 7 \times 6 = 264\,\text{m}^2$

Total surface area of a closed cylinder with base radius $r$ and height $h$

Formula: $A = 2\pi r(r + h)$

Example Question: What is the total surface area of a closed cylinder with base radius $3\,\text{cm}$ and height $7\,\text{cm}$?

Solution: $A = 2\pi(3)(3 + 7) = 60\pi\,\text{cm}^2$

Volume of a hollow cylinder with outer radius $R$, inner radius $r$, and height $h$

Formula: $V = \pi(R^2 - r^2)h$

Example Question: A hollow iron pipe is $21\,\text{cm}$ long with an external radius of $4\,\text{cm}$ and an internal radius of $3\,\text{cm}$. What is the volume of iron in the pipe?

Solution: $V = \frac{22}{7} \times (4^2 - 3^2) \times 21 = \frac{22}{7} \times 7 \times 21 = 462\,\text{cm}^3$

Volume of a right circular cone with base radius $r$ and height $h$

Formula: $V = \frac{1}{3}\pi r^2 h$

Example Question: What is the volume of a right circular cone with base radius $6\,\text{cm}$ and height $7\,\text{cm}$?

Solution: $V = \frac{1}{3}\pi(6^2)(7) = 84\pi\,\text{cm}^3$

Curved surface area of a right circular cone with base radius $r$ and slant height $l$

Formula: $A = \pi r l$

Example Question: What is the curved surface area of a cone with radius $3\,\text{cm}$ and slant height $5\,\text{cm}$?

Solution: $A = \pi(3)(5) = 15\pi\,\text{cm}^2$

Volume of a hemisphere with radius $r$

Formula: $V = \frac{2}{3}\pi r^3$

Example Question: What is the volume of a solid hemisphere with radius $3\,\text{cm}$?

Solution: $V = \frac{2}{3}\pi(3^3) = 18\pi\,\text{cm}^3$

Total surface area of a solid hemisphere with radius $r$

Formula: $A = 3\pi r^2$

Example Question: What is the total surface area of a solid hemisphere with radius $7\,\text{cm}$?

Solution: $A = 3 \times \frac{22}{7} \times 7^2 = 462\,\text{cm}^2$