Mensuration SS3 Cheatsheet

Cheatsheet Content

1. JAMB Questions on Two-Dimensional Shapes 1.1 Triangles Question 1: The sides of a triangle are 6 cm, 8 cm, and 10 cm. Find its area. A) $24 \text{ cm}^2$ B) $30 \text{ cm}^2$ C) $48 \text{ cm}^2$ D) $60 \text{ cm}^2$ Solution Hint: Check if it's a right-angled triangle using Pythagoras theorem. If so, use $A = \frac{1}{2}bh$. (Answer: A) Question 2: An equilateral triangle has a perimeter of 36 cm. Calculate its area. A) $36\sqrt{3} \text{ cm}^2$ B) $72\sqrt{3} \text{ cm}^2$ C) $108 \text{ cm}^2$ D) $18\sqrt{3} \text{ cm}^2$ Solution Hint: Find side length from perimeter, then use $A = \frac{\sqrt{3}}{4}a^2$. (Answer: A) Question 3: The area of a triangle is $48 \text{ cm}^2$. If its base is 12 cm, find its height. A) 4 cm B) 6 cm C) 8 cm D) 10 cm Solution Hint: Use $A = \frac{1}{2}bh$ and solve for $h$. (Answer: C) 1.2 Quadrilaterals Question 4: A rectangular field is 50 m long and 30 m wide. What is its perimeter? A) 80 m B) 100 m C) 160 m D) 1500 m Solution Hint: Use $P = 2(l+b)$. (Answer: C) Question 5: The area of a square is $144 \text{ cm}^2$. Find the length of its diagonal. A) 12 cm B) $12\sqrt{2} \text{ cm}$ C) 24 cm D) $6\sqrt{2} \text{ cm}$ Solution Hint: Find side length from area ($s^2=A$), then use $d = s\sqrt{2}$. (Answer: B) Question 6: The parallel sides of a trapezium are 7 cm and 13 cm. If its height is 5 cm, calculate its area. A) $50 \text{ cm}^2$ B) $65 \text{ cm}^2$ C) $32.5 \text{ cm}^2$ D) $100 \text{ cm}^2$ Solution Hint: Use $A = \frac{1}{2}(a+b)h$. (Answer: A) Question 7: A rhombus has diagonals of length 16 cm and 12 cm. Find its area. A) $48 \text{ cm}^2$ B) $96 \text{ cm}^2$ C) $192 \text{ cm}^2$ D) $28 \text{ cm}^2$ Solution Hint: Use $A = \frac{1}{2}d_1d_2$. (Answer: B) 1.3 Circles Question 8: A circle has a radius of 14 cm. Calculate its circumference. (Take $\pi = \frac{22}{7}$) A) 22 cm B) 44 cm C) 88 cm D) 154 cm Solution Hint: Use $C = 2\pi r$. (Answer: C) Question 9: Find the area of a circle whose diameter is 28 cm. (Take $\pi = \frac{22}{7}$) A) $154 \text{ cm}^2$ B) $308 \text{ cm}^2$ C) $616 \text{ cm}^2$ D) $1232 \text{ cm}^2$ Solution Hint: Find radius ($r=d/2$), then use $A = \pi r^2$. (Answer: C) Question 10: The perimeter of a sector of a circle with radius 10 cm is 32 cm. Find the angle of the sector in degrees. (Take $\pi = \frac{22}{7}$) A) $36^\circ$ B) $72^\circ$ C) $108^\circ$ D) $144^\circ$ Solution Hint: Perimeter = Arc length + $2r$. Find arc length, then use $L = \frac{\theta}{360}2\pi r$. (Answer: B) Question 11: An arc of a circle subtends an angle of $60^\circ$ at the centre. If the radius is 3 cm, calculate the length of the arc. (Take $\pi = \frac{22}{7}$) A) $\frac{11}{7} \text{ cm}$ B) $\frac{22}{7} \text{ cm}$ C) $\frac{33}{7} \text{ cm}$ D) $\frac{44}{7} \text{ cm}$ Solution Hint: Use $L = \frac{\theta}{360}2\pi r$. (Answer: B) 2. JAMB Questions on Three-Dimensional Shapes 2.1 Cuboids and Cubes Question 12: A rectangular tank has dimensions 10 m by 8 m by 5 m. What is its volume? A) $23 \text{ m}^3$ B) $400 \text{ m}^3$ C) $130 \text{ m}^3$ D) $80 \text{ m}^3$ Solution Hint: Use $V = lbh$. (Answer: B) Question 13: The total surface area of a cube is $294 \text{ cm}^2$. Find the length of one side of the cube. A) 6 cm B) 7 cm C) 8 cm D) 9 cm Solution Hint: Use $A = 6s^2$ and solve for $s$. (Answer: B) 2.2 Cylinders Question 14: A cylindrical can has a radius of 3 cm and a height of 10 cm. Calculate its volume. (Take $\pi = 3.142$) A) $94.26 \text{ cm}^3$ B) $188.52 \text{ cm}^3$ C) $282.78 \text{ cm}^3$ D) $314.2 \text{ cm}^3$ Solution Hint: Use $V = \pi r^2 h$. (Answer: C) Question 15: The curved surface area of a cylinder is $176 \text{ cm}^2$. If its radius is 7 cm, find its height. (Take $\pi = \frac{22}{7}$) A) 2 cm B) 3 cm C) 4 cm D) 5 cm Solution Hint: Use CSA $= 2\pi rh$ and solve for $h$. (Answer: C) 2.3 Cones Question 16: A cone has a base radius of 7 cm and a height of 24 cm. Calculate its volume. (Take $\pi = \frac{22}{7}$) A) $1232 \text{ cm}^3$ B) $1642 \text{ cm}^3$ C) $1848 \text{ cm}^3$ D) $3696 \text{ cm}^3$ Solution Hint: Use $V = \frac{1}{3}\pi r^2 h$. (Answer: A) Question 17: The height of a cone is 8 cm and its base radius is 6 cm. Find its slant height. A) 7 cm B) 10 cm C) 14 cm D) 100 cm Solution Hint: Use $l = \sqrt{r^2+h^2}$. (Answer: B) Question 18: What is the curved surface area of the cone in Question 17? (Take $\pi = 3.142$) A) $60\pi \text{ cm}^2$ B) $188.52 \text{ cm}^2$ C) $120\pi \text{ cm}^2$ D) $376.99 \text{ cm}^2$ Solution Hint: Use CSA $= \pi rl$. (Answer: B) 2.4 Spheres Question 19: A sphere has a radius of 6 cm. Calculate its volume. (Take $\pi = 3.142$) A) $452.45 \text{ cm}^3$ B) $904.92 \text{ cm}^3$ C) $113.10 \text{ cm}^3$ D) $226.19 \text{ cm}^3$ Solution Hint: Use $V = \frac{4}{3}\pi r^3$. (Answer: B) Question 20: What is the surface area of the sphere in Question 19? (Take $\pi = 3.142$) A) $452.45 \text{ cm}^2$ B) $113.10 \text{ cm}^2$ C) $226.19 \text{ cm}^2$ D) $904.92 \text{ cm}^2$ Solution Hint: Use $A = 4\pi r^2$. (Answer: A) Question 21: A solid hemisphere has a total surface area of $147\pi \text{ cm}^2$. Find its radius. A) 3 cm B) 5 cm C) 7 cm D) 9 cm Solution Hint: Use TSA $= 3\pi r^2$ for a solid hemisphere. (Answer: C) 3. JAMB Questions on Similar Shapes Question 22: Two similar cones have their heights in the ratio 2:3. If the volume of the smaller cone is $24 \text{ cm}^3$, what is the volume of the larger cone? A) $36 \text{ cm}^3$ B) $54 \text{ cm}^3$ C) $81 \text{ cm}^3$ D) $108 \text{ cm}^3$ Solution Hint: Ratio of volumes is $k^3$. $k = \frac{3}{2}$. (Answer: C) Question 23: The ratio of the areas of two similar triangles is 4:9. If the perimeter of the larger triangle is 30 cm, what is the perimeter of the smaller triangle? A) 10 cm B) 15 cm C) 20 cm D) 25 cm Solution Hint: Ratio of areas is $k^2$. Find $k$, then use ratio of perimeters is $k$. (Answer: C) 4. JAMB Questions on Composite Solids Question 24: A solid consists of a cylinder with a hemisphere mounted on top. The cylinder has a height of 10 cm and a radius of 3 cm. Calculate the total volume of the solid. (Take $\pi = 3.142$) A) $282.78 \text{ cm}^3$ B) $348.65 \text{ cm}^3$ C) $314.16 \text{ cm}^3$ D) $376.99 \text{ cm}^3$ Solution Hint: Volume = Volume of cylinder + Volume of hemisphere. $V_{cyl} = \pi r^2 h$, $V_{hemi} = \frac{2}{3}\pi r^3$. (Answer: B)