Geometry & Measurement Concepts Flashcards

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

Answer: (76π/3) cm³

V = (πh/3)(R²+Rr+r²) = (4π/3)(25+10+4) = (4π/3)·39 = 52π cm³.

In a triangle, the sides are 7, 8, and 9 cm. Using Heron's formula, what is the area?

Answer: 12√5 cm²

s = 12; A = √(12·5·4·3) = √720 = 12√5 cm².

A rectangle has its length increased by 20% and width decreased by 10%. By what percentage does the area change?

Answer: 8% increase

New area = 1.2 × 0.9 × original = 1.08 × original, an 8% increase.

Three mutually tangent circles each have radius 1. What is the area of the curved triangle between them (the gap region)?

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.

A quadrilateral has vertices (0,0), (4,0), (4,3), (0,5). What is its area?

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.

An isosceles trapezoid has parallel sides of 6 cm and 10 cm, and legs of 5 cm each. What is its area?

Answer: 8√21 cm²

Each leg projects 2 cm horizontally; h = √(25−4) = √21; area = ½(6+10)·√21 = 8√21 cm².

A square pyramid has base side 6 cm and height 4 cm. What is the total surface area?

Answer: 96 cm²

Slant height = √(4²+3²) = 5; lateral area = 4·½·6·5 = 60; base = 36; total = 96 cm².