PSAT 8/9 Math · Lesson 4 of 4
Geometry: lines, triangles, area, and volume
Label a diagram before choosing a formula, and keep perimeter, area, and volume distinct.
Use the information that is actually given
A sketch is not a measurement. If a problem says an angle is 70°, write 70 on that angle. If it does not say the triangle is equilateral, do not assume the sides are equal because they look similar.
The sum of angles in a triangle is 180°. A straight line is 180°. Vertical angles are equal. Use those facts to fill missing angles before you try a side formula.
Match the formula to the quantity
Perimeter is a length, so it adds lengths. Area is measured in square units. For a triangle, area is one-half times base times the perpendicular height, not half of every side. Volume uses cubic units and a height that is perpendicular to the base.
If one side of a right triangle is missing, the Pythagorean theorem relates the legs a and b to the hypotenuse c by a² + b² = c². The hypotenuse is the side opposite the right angle, and it is the longest side.
Worked example
A right triangle has legs 6 and 8. What is the area, and what is the hypotenuse?
- The legs are perpendicular, so one can be the base and the other the height. Area = (1/2) × 6 × 8 = 24.
- The hypotenuse is √(6² + 8²) = √(36 + 64) = √100 = 10.
- The area uses the legs. The perimeter would be 6 + 8 + 10 = 24, which happens to equal the area numerically but is a length, not an area.
Why this works. Ask whether the requested quantity is a length, an area, or a volume before you calculate.
Check your understanding
- Label given lengths and angles on the diagram.
- Choose a formula that matches perimeter, area, or volume.
- Use perpendicular height for a triangle, and the longest side as the hypotenuse of a right triangle.