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SAT Math Unit 7: Lines, angles, triangles, area & volume

Geometry questions use a handful of facts repeatedly. Many formulas are on the SAT reference sheet; knowing them cold saves time.

Covers: Angle relationships, triangle angle sum, similar triangles, area and perimeter, volume of solids

Study guide

Angles on a straight line add to 180°, angles around a point add to 360°, and vertical angles (opposite angles where two lines cross) are equal. When a transversal crosses parallel lines, corresponding and alternate interior angles are equal and same-side interior angles add to 180°. The three angles of any triangle add to 180°, and an exterior angle equals the sum of the two remote interior angles.

Similar triangles have equal angles and proportional sides. Two pairs of equal angles are enough to prove similarity. If the side ratio is a:b, the area ratio is a²:b² and the volume ratio of similar solids is a³:b³.

Area and perimeter formulas: rectangle lw, triangle ½bh, parallelogram bh, trapezoid ½(b₁ + b₂)h, circle πr² with circumference 2πr. For composite figures, add or subtract simpler shapes.

Volume formulas: a rectangular prism is lwh, a cylinder is πr²h, a cone is ⅓πr²h, a pyramid is ⅓Bh, and a sphere is (4/3)πr³. The SAT reference sheet lists these, but knowing them saves time. Read carefully for radius versus diameter.

Surface area is the total area of every outside face. For a rectangular prism it is 2(lw + lh + wh). For a composite solid, add the faces you can see and leave out any faces that are glued together or hidden. In an isosceles triangle the angles opposite the equal sides are equal. An equilateral triangle has three 60° angles and area (√3/4)s².

Any side of a triangle must be shorter than the sum of the other two sides and longer than their difference.

Key ideas

• Angles on a line sum to 180°; vertical angles are equal.
• Triangle angles sum to 180°.
• Similar triangles: side ratio k, area ratio k², volume ratio k³.
• Parallel lines with a transversal: corresponding and alternate interior angles are equal.
• Volume formulas: prism Bh, cylinder πr²h, cone ⅓πr²h, sphere (4/3)πr³.

Formulas and rules

Common mistakes

• Using the side ratio for an area ratio.
• Using slant height instead of height for a cone.

Words to know

Sum of angles on a straight line

180°

Sum of a triangle's angles

180°

Area ratio of similar figures

The square of the side ratio (2:3 → 4:9).

Volume of a cylinder

πr²h

Volume of a cone

⅓πr²h

Vertical angles

Opposite angles where two lines cross; equal.

Exterior angle theorem

An exterior angle equals the sum of the two remote interior angles.

Similar triangles

Equal angles and proportional sides.

Area of a triangle

½bh

Volume of a sphere

(4/3)πr³

Volume of a pyramid

⅓Bh

Sample practice questions

247 more questions like these, from level 1 to 5, are in the practice set.