TMUA Geometry & Measurement Concepts 4

Question 1

A frustum (truncated cone) has top radius 2 cm, base radius 5 cm, and height 4 cm. What is its volume?

Accepted Answer

(76π/3) cm³

Answer

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?

Accepted Answer

12√5 cm²

Answer

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?

Accepted Answer

8% increase

Answer

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)?

Accepted Answer

(√3 - π/2) units²

Answer

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?

Accepted Answer

17 units²

Answer

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.

(TMUA) Test of Mathematics for University Admission Practice Test