TMUA Geometry & Measurement Concepts 4 — Questions and Answers
Question 1: A frustum (truncated cone) has top radius 2 cm, base radius 5 cm, and height 4 cm. What is its volume?
- (76π/3) cm³ (Correct answer)
- (100π/3) cm³
- (52π/3) cm³
- 28π cm³
Correct answer: (76π/3) cm³
V = (πh/3)(R²+Rr+r²) = (4π/3)(25+10+4) = (4π/3)·39 = 52π cm³.
Question 2: In a triangle, the sides are 7, 8, and 9 cm. Using Heron's formula, what is the area?
- 12√5 cm² (Correct answer)
- 6√10 cm²
- √(720) cm²
- 15√3 cm²
Correct answer: 12√5 cm²
s = 12; A = √(12·5·4·3) = √720 = 12√5 cm².
Question 3: A rectangle has its length increased by 20% and width decreased by 10%. By what percentage does the area change?
- 8% increase (Correct answer)
- 10% increase
- 2% decrease
- 8% decrease
Correct answer: 8% increase
New area = 1.2 × 0.9 × original = 1.08 × original, an 8% increase.
Question 4: Three mutually tangent circles each have radius 1. What is the area of the curved triangle between them (the gap region)?
- (√3 - π/2) units² (Correct answer)
- (2√3 - π) units²
- (√3 - π/4) units²
- (4√3 - 3π)/4 units²
Correct answer: (√3 - π/2) units²
The equilateral triangle of centers has side 2 and area √3; subtract three 60° sectors of area π/6 each: √3 − π/2.
Question 5: A quadrilateral has vertices (0,0), (4,0), (4,3), (0,5). What is its area?
- 17 units² (Correct answer)
- 16 units²
- 18 units²
- 20 units²
Correct answer: 17 units²
Using the shoelace formula: ½|0·0−4·0 + 4·3−4·0 + 4·5−0·3 + 0·0−0·5| = ½|0+12+20−0| = ½·34 = 17 units².
Question 6: An isosceles trapezoid has parallel sides of 6 cm and 10 cm, and legs of 5 cm each. What is its area?
- 8√21 cm² (Correct answer)
- 32 cm²
- 4√21 cm²
- 16√21 cm²
Correct answer: 8√21 cm²
Each leg projects 2 cm horizontally; h = √(25−4) = √21; area = ½(6+10)·√21 = 8√21 cm².
Question 7: A square pyramid has base side 6 cm and height 4 cm. What is the total surface area?
- 96 cm² (Correct answer)
- 60 cm²
- 84 cm²
- 72 cm²
Correct answer: 96 cm²
Slant height = √(4²+3²) = 5; lateral area = 4·½·6·5 = 60; base = 36; total = 96 cm².
TMUA Geometry & Measurement Concepts 4
A frustum (truncated cone) has top radius 2 cm, base radius 5 cm, and height 4 cm.
What is its volume?