A triangle has a base of 10 cm. The slant side adjacent to the base measures 8 cm, and the perpendicular height from the base to the opposite vertex is 6 cm. What is the area?
A80 cm² — multiply base times slant side
B40 cm² — use (1/2) × base × slant side
C30 cm² — use (1/2) × base × perpendicular height
D60 cm² — multiply base times perpendicular height
The formula A = (1/2)bh requires the *perpendicular* height h — the straight-line distance from the base to the opposite vertex, measured at a right angle. Here h = 6 cm, so A = (1/2)(10)(6) = 30 cm². Using the slant side (8 cm) instead of the height is the most common error with non-right triangles. The slant side is a side of the triangle, not its height.
Question 2 Multiple Choice
A triangle has three base-height pairs that can be used to compute its area: (base 5, height 12) and (base 10, height 6). What must the height be when the third side of length 13 is used as the base?
A13 — the height equals the base when a different side is chosen
B≈ 4.6 — because all three base-height pairs must yield the same area
CCannot be determined without knowing the triangle's angles
D6 — the height is fixed regardless of which base you choose
All three base-height pairs give the same area because there is only one area for a given triangle. Using (base 5, height 12): A = (1/2)(5)(12) = 30. Using (base 10, height 6): A = (1/2)(10)(6) = 30. So (1/2)(13)(h) = 30, giving h = 60/13 ≈ 4.6. This confirms that you can choose any side as the base as long as you use the *corresponding* perpendicular height.
Question 3 True / False
The height of a triangle is typically one of its three sides.
TTrue
FFalse
Answer: False
The height (altitude) of a triangle is the perpendicular distance from a chosen base to the opposite vertex — it is not a side of the triangle unless the triangle is a right triangle and you use the legs as base and height. For non-right triangles, the height is a separate segment drawn from the vertex perpendicular to the base (or the extended base). Using a slant side as the height is the most common error in computing triangle area.
Question 4 True / False
If you double the height of a triangle while keeping the base the same, the area doubles.
TTrue
FFalse
Answer: True
A = (1/2)bh. If h becomes 2h, the new area is (1/2)b(2h) = 2 × (1/2)bh = 2A. Area is directly proportional to height (and to base). This is a direct consequence of the formula and is useful for scaling: doubling any one dimension while holding the other constant doubles the area.
Question 5 Short Answer
Why does the formula A = (1/2)bh for the area of a triangle include the factor of 1/2?
Think about your answer, then reveal below.
Model answer: Because every triangle is exactly half of a parallelogram (or rectangle) with the same base and height. If you duplicate a triangle and rotate the copy 180°, the two triangles fit together to form a rectangle (for right triangles) or parallelogram (for any triangle). That parallelogram has area b × h, so each triangle is half: (1/2)bh.
This geometric reasoning makes the formula unforgettable once understood. It also explains why the height must be perpendicular: the area of the parallelogram is base times *perpendicular* height, so halving it gives (1/2) × base × perpendicular height. Using a slant dimension would give the wrong parallelogram area and thus the wrong triangle area.