Geometry Practice Test: Area, Volume and Coordinate Geometry

Trapezoid area = ½ · (b₁ + b₂) · h = ½ · 22 · 5 = 55 cm². The value 110 cm² forgets the factor ½.

A regular hexagon is made of 6 equilateral triangles with side 4, each with area (3/4) · 4² = 43. The total is 6 · 43 = 243 ≈ 41.6.

Volume = πr²h = π · 9 · 10 = 90π m³. The value 30π m³ is the volume of a cone with the same base and height, and 60π m³ is the lateral area 2πrh.

Volume = ⅓πr²h = ⅓ · π · 36 · 8 = 96π in³. Forgetting the ⅓ gives 288π in³, and 60π in² is the lateral surface area πrℓ with slant height 10.

Volume = (4/3)πr³ = (4/3) · π · 27 = 36π cm³. The value 108π cm³ forgets the division by 3.

Surface area = 2(lw + lh + wh) = 2(12 + 20 + 15) = 94 ft². The value 60 is the volume (in ft³), and 47 counts each face only once.

Tripling every dimension multiplies the volume by 3³ = 27, so the new volume is 40 · 27 = 1,080 cm³. Tripling only once gives 120 cm³, and 360 cm³ uses 3² = 9.

The horizontal change is 6 and the vertical change is 8, so the distance is 6² + 8² = 100 = 10. The value 14 adds the two changes instead of using the Pythagorean theorem.

Average the coordinates: ((−2 + 6)/2, (3 + (−7))/2) = (2, −2). The point (4, −4) adds the coordinates without dividing by 2, and (8, −10) is the difference between the endpoints.

Solving for y gives y = −(2/3)x + 7/3, so line k has slope −2/3. A perpendicular slope is the opposite reciprocal: 3/2. The slope −3/2 is the reciprocal without the sign change.

P is one third of the way from A to B: (1 + ⅓ · 9, 2 + ⅓ · 12) = (4, 6). The point (7, 10) is two thirds of the way, which is the ratio 2 : 1, and (5.5, 8) is the midpoint.

The base along the x-axis is 8 and the height is the y-coordinate of the third vertex, 6. Area = ½ · 8 · 6 = 24; forgetting the ½ gives 48.

Slopes: AB and CD are 1/2, BC and DA are −2, so opposite sides are parallel and adjacent sides are perpendicular (1/2 · (−2) = −1). But AB = 20 and BC = 5, so the sides are not all equal: a rectangle, not a square.

The semicircle has diameter 6, so radius 3 and area ½ · π · 3² = 4.5π ≈ 14.1 ft². Adding the rectangle gives 60 + 14.1 = 74.1 ft²; 88.3 ft² uses a full circle instead of a half.

Pyramid volume = ⅓ · base area · height = ⅓ · 36 · 10 = 120 m³. The value 360 m³ is the volume of a prism with the same base and height, and 180 m³ uses ½ instead of ⅓.