Cross Sections
What 2D shape is formed by slicing a cube straight through the parallel to the base?
Square
A town has 2,000 people and covers an area of 10 km². What is the population density?
2000÷10=200 people/km²
A trapezoid has bases 12 m and 8 m with a height of 5m. Find the area.
Trapezoid = 1/2(12+8)(5)= 50m^2
In a circle with radius 10, what is the circumference?
Circumference = 2πr= 2π(10) = 20π or 62.83
A circle has radius 6. Find the area of a 90° sector.
Area of Sector = 90/360(π⋅6²) = 9π≈ 28.27
What shape could result from slicing a cone perpendicular to the base?
Triangle
A wildlife reserve is 50 km² with 25,000 animals. What is the population density?
25,000÷50=500 animals/km²
Find the area of a rectangle (8 × 12) with an isosceles triangle on top (base = 8, height = 6).
Rectangle = 8×12=96;
triangle = 1/2(8)(6)=24
Total = 120 m²
In a circle with radius 7, what is the arc length for a 90° angle?
Arc length = 90/360(2π⋅7) = 3.5π≈10.99
A circle has radius 10. Find the area of a 60° sector.
Area of Sector = 60/360(π⋅10²) = 16.66π≈ 52.35
A composite solid is made of a cone attached to a rectangular prism. Which of these shapes could NOT be a cross-section of the solid: triangle, rectangle, trapezoid, or hexagon?
hexagon
A rectangular park measures 6 km by 4 km. It has 12,000 people visiting. What is the density per km²?
Area = 6×4=24;
Density = 12,000÷24=500 people/km²
A figure is composed of a square (side = 10 m) and a semicircle (diameter = 10 m). Find the total area.
Square = 10×10=100;
semicircle = 1/2π(5²)=12.5π≈39.3;
Total ≈ 139.3m²
In a circle with radius 12, find the arc length for a 60° central angle.
Arc length = 60/360(2π⋅12) = 4π≈ 12.56
A circle has radius 12. Find the area of a 150° sector.
Area of Sector = 150/360(π⋅12²) = 60π≈ 188.49
A square pyramid is sliced parallel to its base at half its height. What shape is the cross-section?
Square
A triangular region has legs 8 km and 10 km. Population is 20,000. What is density?
Area = 1/2(8)(10)=40;
Density = 20,000÷40=500 people/km²
A shape is made from a rectangle (8 × 5 m) with two semicircles attached on the shorter sides (diameter = 5). Find the total area.
Rectangle = 40;
two semicircles = one full circle = π(2.5²)=6.25π≈19.6
Total ≈ 59.6 m²
In a circle with diameter 20, find the arc length for a 135° central angle.
Arc length = 135/360(2π⋅10) = 7.5π≈ 23.56
A circle has diameter of 14. Find the area of a 45° sector.
Area of Sector = 45/360(π⋅7²) = 6.125π≈ 19.24
A house-shaped solid is formed from a rectangular prism and a triangular prism on top. If you slice vertically through the center, what cross-section shape appears?
House/ Pentagon
A circular island has a radius of 7 km. If 154,000 people live there, what is the approximate density per km²?
Area = πr²=π(7)²=154 km²;
Density = 154,000÷154=1,000 people/km²
A composite figure is made from a rectangle (10 × 8 m), a right triangle (base 10 m, height 6 m) on top, and a semicircle with radius 4 m attached on one side of the rectangle. Find the total area.
Rectangle = 80;
triangle = 1/2(10)(6)=30;
semicircle = 1/2π(4²)=8π≈25.1;
Total ≈ 135.1 m²
A circle has diameter 15. Find the arc length for a 225° central angle.
Arc length = 225/360(2π⋅7.5) = 9.375π≈ 29.45
A circle has diameter of 20. Find the area of a 225° sector.
Area of Sector = 225/360(π⋅10²) = 62.5π≈ 196.34