Area and Volume
What is the area of a triangle with a base of 10 units and a height of 6 units? (A = ½ × base × height)
Area = ½ × 10 × 6 = 30 units²
Two sides of a rectangle have lengths (x + 2) and (2x – 1). If the perimeter is 30 units, set up the system of equations to find x.
2(x + 2) + 2(2x – 1) = 30
Simplify: (x²)(x³). What geometric quantity could this represent in volume terms?
(x²)(x³) = x⁵; This could represent volume if x is a unit of length
Factor: x² + 5x + 6. This could represent the area of what type of figure?
(x + 2)(x + 3); Could represent a rectangle with those side lengths
What is the definition of tangent in a right triangle? (Use SOHCAHTOA)
Tangent = Opposite / Adjacent
Find the volume of a rectangular prism with dimensions 3 cm × 4 cm × 5 cm. (V = l × w × h)
Volume = 3 × 4 × 5 = 60 cm³
The base and height of a triangle are (2x + 1) and (x + 3), and the area is 30. Write the system of equations.
(2x + 1)(x + 3) = 60
A cube has volume x⁶. What is the side length of the cube? (Volume of cube = s³)
If x⁶ = s³, then s = x² (side length of the cube is x²)
The area of a rectangle is x² – 9. Factor the expression. What are the possible dimensions?
(x + 3)(x – 3); Dimensions could be x + 3 and x – 3
A right triangle has an angle of 30°. What is the sine of 30°? (Use a calculator or knowledge of special angles)
sin(30°) = 1/2
A circle has a radius of 7 units. What is its area? (Use A = πr², π ≈ 3.14)
Area = πr² = 3.14 × 49 = 153.86 units²
A square and a rectangle have equal areas. The square’s side is (x), and the rectangle’s dimensions are (x + 2) and (x – 1). Set up and solve the system.
Square: x²; Rectangle: (x + 2)(x – 1) = x² + x – 2;
So, x² = x² + x – 2 → x – 2 = 0
x = 2
Area = 4
Simplify the expression for the area of a square with side (2x²).
Side = 2x²; Area = (2x²)² = 4x⁴
A square has area 4x² – 12x + 9. Factor the area to find the side length.
(2x – 3)²; So side length = 2x – 3
In a right triangle, the adjacent side is 4 units, and the hypotenuse is 5 units. What is cos(θ)?
cos(θ) = Adjacent / Hypotenuse = 4/5
A cylinder has a radius of 3 cm and a height of 10 cm. What is its volume? (V = πr²h)
Volume = πr²h = 3.14 × 9 × 10 = 282.6 cm³
The sum of the length and width of a rectangle is 14 units. The length is twice the width. What are the dimensions?
Let width = x, then length = 2x. x + 2x = 14 → x = 14/3 (not an integer, approx. 4.67 units)
Simplify: (3x²y)(2xy³).
6x³y⁴
Factor: x² – 16. How could this represent a difference in squared lengths?
(x + 4)(x – 4); Represents a difference in squared lengths (e.g., area of larger square – smaller square)
A ladder leans against a wall making a 60° angle with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Use sin(60°))
sin(60°) = Height / Ladder Length
So, Height = 10 * sin(60°)
Height ≈ 10 * 0.866 ≈ 8.66 ft
A composite figure includes a rectangle (10 × 4) and a semicircle on top with a diameter of 10. What is the total area of the figure? (Use A = lw + ½πr²)
Rectangle area = 10 × 4 = 40; Semicircle area = ½ × π × (5)² = 39.25; Total ≈ 79.25 units²
A cone and a cylinder have equal volumes. The cone has height (x) and radius 3. The cylinder has height 6 and radius x. Set up a system to find x. (Cone V = ⅓πr²h, Cylinder V = πr²h)
Cone: V = (1/3)π(3)²(x) = πx9; Cylinder: V = πx² × 6;
ANSWER = 3πx = 6πx²
Simplify 2x-2 (xy)2
2y2
A triangle’s area is ½(x² + 6x + 8). Factor the expression and interpret the base and height.
(x + 2)(x + 4); Area = ½ × base × height → base = x + 2, height = x + 4
A triangle has sides of 5, 12, and 13 units. Verify whether it is a right triangle, then find tan(θ) where θ is the angle opposite the side of length 5.
First, check if it’s a right triangle:
52+122=25+144=1695^2 + 12^2 = 25 + 144 = 16952+122=25+144=169
132=16913^2 = 169132=169 → Yes, it’s a right triangle.
tan(θ) = Opposite / Adjacent = 5 / 12 ≈ 0.4167