Volumes of solids · Grade 12 · 4 units
In Volumes of solids (Grade 12 · 4 units) you practise 5 skills: The volume of a prism · The volume of a pyramid · The height by vectors · The height by Pythagoras and trigonometry · Bases of other shapes. The volume of a prism is the area of its base times its height. The volume of a pyramid is a third of the area of its base times its height. The height is the length of the segment from the top base of the prism, or from the apex of the pyramid, perpendicular to the plane of the base. In a right prism the height is the lateral edge, and in a right pyramid it comes down to the centre of the base.
Find the area of the base as in earlier years: a rectangle and a parallelogram, a triangle by two sides and the angle between them, a rhombus by its diagonals, a trapezium, an equilateral triangle and a regular hexagon, which is made of six equilateral triangles. When the solid is not right, there are four ways to find its height. The coordinates of both ends of the height are given. The height can be seen, for example a lateral edge perpendicular to the base. Or show that a vector is perpendicular to two vectors of the base that are not collinear. Or the length of the height comes from a right triangle, by Pythagoras, the sine or the tangent. When the base lies in the plane where the third coordinate is zero, the height is the third coordinate of the apex, even when the apex is not above the base.
Tip: Remember the third: without it you get the volume of a prism, not a pyramid.
What you learn
The volume of a prism
For example: The box ABCDA′B′C′D′ has AB = 3 , BC = 7 and AA′ = 11 . What is the volume of the box?
Answer and solution
The answer: 231
The base is a rectangle of area 3 × 7 = 21 , so the volume is 21 × 11 = 231 .
The volume of a pyramid
For example: In the right pyramid SABCD the base is a square of side 4 , and the point O is the centre of the base. The height of the pyramid is SO = 12 . What is the volume of the pyramid?
Answer and solution
The answer: 64
The area of the base is 4² = 16 . So the volume is 16 × 12 ÷ 3 = 64 .
The height by vectors
For example: The base of the pyramid SABC is the triangle with corners A(0, 3, 0) , B(2, 3, 0) and C(0, 7, 0) , and its apex is S(6, 5, 3) . What is the volume of the pyramid?
Answer and solution
The answer: 4
The sides AB = 2 and AC = 4 are perpendicular, so the area of the base is 2 × 4 ÷ 2 = 4 . The height is 3 , and the volume is 4 × 3 ÷ 3 = 4 .
The height by Pythagoras and trigonometry
For example: In the right pyramid SABCD the base is a square of side 18 . The point M is the midpoint of BC , and the height of the side face is SM = 15 . What is the volume of the pyramid?
Answer and solution
The answer: 1296
The point O is the centre of the base, so OM = 9 . In the right triangle SOM the height is SO = √(15² − 9²) = 12 . So the volume is 324 × 12 ÷ 3 = 1296 .
Bases of other shapes
For example: The base of the right prism ABCDA′B′C′D′ is a rhombus with diagonals AC = 4 and BD = 12 . The height of the prism is AA′ = 3 . What is the volume of the prism?
Answer and solution
The answer: 72
The area of a rhombus is half the product of its diagonals: 4 × 12 ÷ 2 = 24 . So the volume is 24 × 3 = 72 .
Not on the site yet
Parts of this topic that are not built yet: The volume of a solid made of two solids.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).